NN 


THRESHOLDS 
OF   SCIENCE 


MATHEMATICS 


C.A.Laisant 


rnia 
,1 


Southern  Branch 
of  the 

University  of  California 

Los  Angeles 

Form  L   I 


L14 


This  book  is  DUK  on  the  last  date  stamped  below 


NOV  6 
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""""• 


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MAR  2  7 

JUL5    1952 

NOV  ^6 1952 
JUN  2  4  ^53 
9     1953 


teC  4     1953 
JUL     9 

KEC'D  MLD 

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OCT  i  2  1382 


• 


-  '  - 


THRESHOLDS  OF  SCIENCE 


GENERAL  FOREWORD 

THERE  are  many  men  and  women  who,  from  lack  of 
opportunity  or  some  other  reason,  have  grown  up  in 
ignorance  of  the  elementary  laws  of  science.  They  feel 
themselves  continually  handicapped  by  this  ignorance. 
Their  critical  faculty  is  eager  to  submit,  alike  old  estab- 
lished beliefs  and  revolutionary  doctrines,  to  the  test  of 
science.  But  they  lack  the  necessary  knowledge. 

Equally  serious  is  the  fact  that  another  generation  is 
at  this  moment  growing  up  to  a  similar  ignorance.  The 
child,  between  the  ages  of  six  and  twelve,  lives  in  a  wonder- 
land of  discovery ;  he  is  for  ever  asking  questions,  seeking 
explanations  of  natural  phenomena.  It  is  because  many 
parents  have  resorted  to  sentimental  evasion  in  their 
replies  to  these  questionings,  and  because  children  are 
often  allowed  either  to  blunder  on  natural  truths  for 
themselves  or  to  remain  unenlightened,  that  there 
exists  the  body  of  men  and  women  already  described. 
On  all  sides  intelligent  people  are  demanding  something 
more  concrete  than  theory  ;  on  all  sides  they  are  turning 
to  science  for  proof  and  guidance. 

To  meet  this  double  need — the  need  of  the  man  who 
would  teach  himself  the  elements  of  science,  and  the 
need  of  the  child  who  shows  himself  every  day  eager  to 
have  them  taught  him — is  the  aim  of  the  "  Thresholds  of 
Science"  series. 

This  series  consists  of  short,  simply  written  monographs 
by  competent  authorities,  dealing  with  every  branch  of 
science — mathematics,  zoology,  chemistry  and  the  like. 
They  are  well  illustrated,  and  issued  at  the  cheapest 
possible  price.  When  they  were  first  published  in  France 
they  met  with  immediate  success,  showing  that  science 

t 


ii  GENERAL  FOREWORD 

is  not  necessarily  dull  or  fenced  off  by  a  barrier  of 
technical  jargon.  Of  course,  specialisation  in  this  as  in 
other  subjects  is  not  for  everyone,  but  the  publication  of 
this  se.-ies  of  books  enables  any  man  or  we  man  to  learn, 
any  child  to  be  taught,  to  pass  with  understanding  and 
safety  the  "  Thresholds  of  Science." 


MATHEMATICS 


THRESHOLDS    OF    SCIENCE 


VOLUMES  ALREADY  PUBLISHED 
ZOOLOGY  by  E.  Brocket. 

BOTANY  by  E.  Brucker. 

CHEMISTRY          by  Georges  Darzens. 
MECHANICS          by  C  E.  Goillaume. 


THRESHOLDS     OF    SCIENCE 


MATHEMATICS 

BY 

C.  A.  LAISANT 


ILLUSTRATED 


DOUBLED  AY,  PAGE  &  CO., 

GARDEN  CITY  NEW  YORK 

1914. 

3?^  /  2* 


L  n 

CONTENTS 


PAGE 

GENERAL  FOREWORD      ......  i 

1.  STROKES        ........  1 

2.  ONE   TO   TEN            .            .             .            .            .            .            .  4 

3.  MATCHES   OR   STICKS  ;     BUNDLES   AND   FAGGOTS            .  5 

4.  ONE   TO   A   HUNDRED      ......  7 

6.   THE   ADDITION   TABLE    ......  9 

6.  SUMS 10 

7.  SUBTRACTION 12 

8.  THOUSANDS   AND   MILLIONS      .....  14 

9.  COLOURED    COUNTERS     .  .  .  .  .  .17 

10.  FIGURES         ........  19 

11.  STICKS   END   TO   END 23 

12.  THE    STRAIGHT   LINE        .  .  .  .  .  .23 

13.  SUBTRACTION   WITH  LITTLE   STICKS  .  .  .25 

14.  WE   BEGIN   ALGEBRA       .  .  .  .  .  .26 

15.  CALCULATIONS  ;    MEASURES  ;    PROPORTIONS        .            .  31 

16.  THE   MULTIPLICATION   TABLE               ....  33 

17.  PRODUCTS 35 

18.  CURIOUS   OPERATIONS 38 

19.  PRIME   NUMBERS   .......  40 

20.  QUOTIENTS  ........  42 

21.  THE   DIVIDED   CAKE  ;    FRACTIONS    ....  45 

22.  WE   START   GEOMETRY    ......  52 

23.  AREAS 60 

24.  THE   ASSES'    BRIDGE 64 

25.  VARIOUS   PUZZLES  ;    MATHEMATICAL  MEDLEY    .            .  66 

26.  THE   CUBE   IN   EIGHT   PIECES               ....  68 

27.  TRIANGULAR  NUMBERS. — THE  FLIGHT  OF  THE  CRANES  70 

28.  SQUARE   NUMBERS 72 

29.  THE   SUM   OF   CUBES 76 

30.  THE  POWERS   OF    11 79 

31.  THE  ARITHMETICAL  TRIANGLE  AND   SQUARE      .            .  80 


viii  CONTENTS 

PAGE 

32.  DIFFERENT.1  WATS   OF   COUNTING      ....         82 

33.  BINARY   NUMERATION 87 

34.  ARITHMETICAL  PROGRESSIONS  ....         89 

35.  GEOMETRIC   PROGRESSIONS       .  .  .  .  .91 

36.  THE   GRAINS   OF   CORN   ON    THE   CHESSBOARD     .  .         93 

37.  A  VERY   CHEAP   HOUSE  .....         94 

38.  THE   INVESTMENT   OF  A   CENTIME     ....         95 

39.  THE   CEREMONIOUS   DINNER 96 

40.  A  HUGE   NUMBER  ......       100 

41.  THE   COMPASS   AND   PROTRACTOR      ....       102 

42.  THE   CIRCLE 104 

43.  THE   AREA   OF   THE   CIRCLE 106 

44.  CRESCENTS   AND   ROSES  .....       107 

45.  SOME  VOLUMES 109 

46.  GRAPHS;    ALGEBRA   WITHOUT   CALCULATIONS    .  .112 

47.  THE  TWO   WALKERS         .  .  .  .  .  .115 

48.  FROM   PARIS   TO   MARSEILLES  .  .  .  .117 

49.  FROM  HAVRE  TO   NEW   YORK  .  .  .  .119 

50.  WHAT  KIND   OF  WEATHER   IT   IS       .  .  .  .       121 

51.  TWO   CYCLISTS   FOR   ONE   MACHINE  .  .  .123 

52.  THE   CARRIAGE   THAT   WAS   TOO   SMALL      .  .  .       125 

53.  THE   DOG   AND   THE   TWO   TRAVELLERS       .  .  .       127 

54.  THE   FALLING   STONE 128 

55.  THE   BALL  TOSSED   UP    .  .  .  .  .  .       130 

56.  UNDERGROUND   TRAINS 132 

57.  ANALYTICAL   GEOMETRY 133 

58.  THE   PARABOLA 136 

59.  THE   ELLIPSE 136 

60.  THE   HYPERBOLA 138 

61.  THE   DIVIDED   SEGMENT 139 

62.  DOH,   ME,   SOH  ;    GEOMETRICAL  HARMONIES        .  .       141 

63.  A  PARADOX  :    64  =  65 143 

64.  MAGIC   SQUARES 145 

65.  FINAL  REMARKS 146 

INDEX 155 


MATHEMATICS 


1.  Strokes. 

ONE  of  the  first  faculties  which  we  should  develop  in 
the  child,  from  the  age  when  his  cerebral  activity  wakens, 
is  that  of  drawing.  Nearly  always,  he  has  the  instinctive 
taste  for  it,  and  we  must  encourage  him  in  it,  long  before 
undertaking  to  teach  him  writing  or  reading. 

With  this  object,  we  should  put  a  slate  or  a  sheet  of 
squared  paper  in  his  hands  as  a  beginning,  and  place  a 
pencil  (when  he  is  cleverer,  a  pen)  between  his  little 
fingers  and  make  him  trace  strokes  at  first ;  not  the 
classical  sloping  strokes,  preparatory  to  sloping  writing, 
but  little  lines  following  the  direction  of  the  lines  on  the 
squared  paper,  and  very  regularly  spaced. 

Drawing  these  lines  first  from  top  to  bottom,  then  after 
some  time  from  left  to  right,  the  pupil  will  thus  make 
vertical  strokes  (Fig.  1)  and  horizontal  strokes  (Fig.  2). 

FIG.  1.— Vertical  strokes. 


I    I    I    I 
FIG.  2. — Horizontal  strokes. 


M. 


2  MATHEMATICS 

Gradually  we  will  teach  him  to  draw  long  or  short 
strokes,  to  put  them  between  the  lines  of  the  squared 
paper,  to  draw  new  ones  from  them  which  are  oblique, 
in  every  possible  direction.  Then  we  will  make  him  form 
figures  composed  of  groups  of  long  or  short  strokes. 
We  will  say  something  about  that  below. 

Later,  we  will  make  him  draw  figures  or  begin  curves, 
either  with  instruments  (ruler,  set  square,  compass)  or 
freehand.  These  exercises,  which  develop  skill  of  hand 
and  straightness  of  eye,  should  never  be  left  off  while 
the  educative  period  lasts.  We  only  speak  of  them  here 
in  so  far  as  they  are  indispensable  for  what  will  come 
after :  but,  even  from  this  point  of  view,  we  must  insist 
on  the  fact  that  they  should  be  suggested  and  never  forced. 
If  they  cease  to  be  a  game,  the  object  will  be  lost.  Let 
the  child  scribble  on  his  slate  and  spoil  some  sheets  of 
paper ;  help  him  with  your  advice,  which  he  will  never 
fail  to  ask ;  but  when  he  has  had  enough,  let  him  do 
something  else.  That  is  a  condition  which  is  absolutely 
necessary  to  develop  the  spirit  of  initiative  in  him,  to  keep 
up  his  natural  curiosity  and  to  avoid  fatigue  and  boredom. 

It  would  need  a  whole  book  to  deal  with  this  first 
teaching  of  drawing,  on  which  I  have  been  obliged  to 
say  a  few  words  ;  others  would  be  needed  on  writing  and 
on  reading,  which  should  only  come  afterwards  and  are 
outside  my  subject.  But  all  these  teachings,  applied  to 
childhood,  should  always  be  inspired  by  the  same  funda- 
mental principle,  that  is,  to  keep  the  appearance  of  games, 
to  respect  the  child's  liberty  and  to  give  him  the  illusion 
(if  it  is  one)  that  he  himself  discovers  the  truths  put  before 
his  eyes.  As  to  the  age  at  which  this  first  mathematical 
initiation  should  commence,  starting  with  that  of  drawing, 
and  then  running  parallel  with  it,  there  is  no  absolute 
rule  to  be  laid  down.  But  we  can  say  that  as  a  rule  it  is 
very  rare  if  a  child  of  three  and  a  half  to  four  years  old 
does  not  already  show  a  taste  for  handling  a  pencil :  and 
I  assert  that  at  ten  or  eleven,  it  should  be  easy  to  have 


STROKES  3 

taught  him  all  the  matters  explained  in  what  follows, 
if  his  brain  is  normally  organised. 

More  than  one  may  find  pleasure,  after  some  years, 
in  taking  up  this  little  book  which  is  not  meant  for  him 
now.  His  mind,  perfected  by  further  studies  and  ready 
at  conscious  reasoning,  will  certainly  find  in  it  matter  for 
useful  reflection. 

To  finish  with  these  generalities  and  not  to  repeat  my- 
self unnecessarily,  I  must  point  out  to  families  and  teachers 
who  are  to  be  my  readers  the  greatest  snare  to  be  avoided 
in  the  first  teaching  of  childhood  :  this  is  the  abuse  of 
exercise  of  the  memory,  still  so  general  in  actual  practice, 
and  so  pernicious.  By  teaching  words  to  a  child  and  mak- 
ing him  repeat  them,  we  deform  his  brain,  we  kill  his 
natural  gifts,  and  bring  up  generations  of  beings  without 
initiative,  without  curiosity  and  without  will,  enfeebled, 
depressed,  and  stuffed  up  with  formulae  which  are  not 
understood. 

If  you  love  your  children,  if  you  love  those  confided 
to  you,  if  you  wish  them  to  become  good  and  strong, 
go  back  to  the  principles  of  the  great  minds  and  hearts 
who  were  named  La  Chalotais,1  Frocbel 2  and  Pestalozzi.3 
If  the  earth  was  peopled  with  reasonable  beings,  these 
benefactors  of  humanity  would  have  their  statues  in  every 
country  of  the  world,  and  their  names  would  be  graven  in 
letters  of  gold  in  every  school. 

1  La  Chalotais,  French  magistrate,  born  at  Rennes  (1701 — 1785), 
author  of  "  Essay  on  National  Education." 

2  Frcebel,  German  pedagogue,  born  at  Oberweissbach  (1782 — 1852), 
founder  of  the  "  Kindergartens." 

8  Pestalozzi,  Swiss  educator,  born  at  Zurich  (1746 — 1827) ;  his 
method  served  as  a  basis  for  Fichte,  as  a  means  of  raising  Germany. 


B  2 


4  MATHEMATICS 

2.  One  to  Ten. 

Once  the  habit  has  begun  of  drawing  strokes  regularly — 
and  quickly  enough — he  will  learn  to  count  them  as  he 
makes  them,  pronouncing  the  names,  one,  two,  three, 
four,  five,  six,  seven,  eight,  nine,  ten,  successively. 

Then  he  will  make  groups  of  strokes,  separating  them 
from  one  another  by  spaces,  and  he  will  have  (Figs.  3  and 
4)  diagrams  which  he  will  read  : — 

FIG.  3. 

I       II       III       I    I    I    I       I    I    I    I    I 
one       two         three  four  five 


eight 


ten 


FIG.  4. 

one    two  three  four  five     six  seven  eight  nine  ten 


One,  two,  ...  ten  vertical  strokes,  for  Fig.  3 ; 
One,  two,  .  .  .  ten  horizontal  strokes,  for  Fig.  4. 


MATCHES  OR    STICKS,    ETC.  5 

Then  he  will  put  down  groups  of  beans,  of  grains  of 
corn,  of  counters,  and  of  any  other  things,  and  they  are 
to  be  counted  : — 

One,  two,  .  .  .  ten  beans,  grains  of  corn,  etc. 

We  will  now  suppose  that  these  objects  are  replaced 
by  sheep,  dogs,  men,  etc.,  and  after  these  exercises  are 
repeated  often  enough  and  are  familiar  to  the  child,  we 
can  tell  him  that  the  expressions  he  uses,  for  instance, 
three  strokes,  six  grains  of  corn,  eight  sheep,  are  called 
concrete  numbers. 

Having  considered  a  group  of  five  strokes,  another  of 
five  beans,  another  of  five  counters,  having  imagined 
another  of  five  dogs  or  five  trees,  we  will  tell  him  that  in 
these  different  cases  he  is  always  saying  the  same  word 
five;  we  will  tell  that  this  word,  without  anything  else, 
represents  what  is  called  an  abstract  number,  and  that  he 
can  use  it  to  denote  any  other  group  of  five  things ; 
donkeys,  horses,  houses,  etc. 

It  will  not  be  long  before  the  child  can  count  without  any 
hesitation  from  one  to  ten  in  any  things  at  all.  It  will 
be  well  also  to  accustom  him  as  soon  as  possible  to  grasp 
at  a  glance  the  number  of  things  shown  to  him  quickly, 
without  having  to  count  them  one  by  one :  to  do  this, 
he  must  start  with  very  small  numbers  and  go  on 
gradually. 

3.  Matches  or  Sticks;  Bundles  and  Faggots. 

Beyond  the  different  things  just  mentioned,  to  help  the 
child  to  understand  the  idea  of  concrete  numbers,  which 
can  be  infinitely  varied,  there  are  others  which  we  can 
hardly  recommend  too  highly,  whose  use  is,  in  our  opinion, 
indispensable.  These  are  little  wooden  sticks,  exactly 
like  ordinary  wood  matches  except  that  they  have  no 
inflammable  chemical  preparation.  We  will  sometimes 
call  them  matches,  because  of  this  resemblance,  and  these 


6  MATHEMATICS 

matches — which  do  not  strike — can  be  considered  models 
of  the  strokes  drawn  on  the  slate  or  in  the  copy-book. 
They  should  all  be  the  same  length. 

Having  a  heap  of  these  sticks  before  him,  and  knowing 
quite  well  how  to  count  up  to  ten,  the  child  will  put  aside 
ten,  one  after  the  other,  and  will  put  them  together  in  a 
very  regular  little  bundle,  and  surround  them  with  one 
of  those  little  rubber  bands  which  are  so  convenient  and 
so  widely  used. 

We  will  show  him  then  that  this  bundle  containing  ten 
sticks  can  be  called  a  "  ten  "  of  sticks. 

Then  he  will  make  up  a  fair  number  of  similar  bundles. 
We  will  see  that  he  has  not  made  a  mistake  :  if  he  has  we 
will  make  him  put  it  right. 

Then  showing  him  two  bundles,  we  will  tell  him  that 
in  these  two  bundles  taken  together,  the  number  of  sticks 
which  we  will  show  him  by  untying  the  bundles  and  tying 
them  up  again  is  called  twenty,  and  that  thus  : — 

One  bundle  is  ten  sticks, 
Two  bundles  are  twenty  sticks. 

Then  taking  three,  four,  .  .  .  nine  bundles,  and  doing 
the  same  thing,  we  will  show  him  that 

Three  bundles  are  thirty     sticks 

Four  „  „  forty  „ 

Five  „  „  fifty 

Six  „  „  sixty  „ 

Seven  „  „  seventy     „ 

Eight  „  „  eighty 

Nine  „  „  ninety  „ 

Having  learnt  all  this,  to  conclude  we  will  take  ten 
bundles,  and  we  will  put  them  together  by  a  larger  rubber 
band,  which  will  give  us  a  faggot.  We  will  then  explain 
that  a  faggot  is  a  hundred  sticks,  that  the  number  of  sticks 
in  a  faggot  is  called  a  hundred :  he  will  verify  that  as  ten 
bundles  make  a  faggot,  ten  tens  are  a  hundred. 


ONE  TO  A  HUNDRED  7 

4.  One  to  a  Hundred. 

Taking  a  handful  of  sticks  at  random  (less  than  a 
hundred)  we  will  tell  the  child  that  we  are  going  to  count 
them  with  him.  To  do  this,  he  must  make  bundles,  as 
many  as  possible ;  he  will  come  to  a  point  at  which  he 
will  not  have  enough  sticks  left  to  make  a  bundle.  Then, 
putting  all  the  made-up  bundles  at  his  left,  and  the  sticks 
left  over  at  his  right,  we  will  make  him  say  the  two 
numbers  separately ;  then,  putting  them  together,  he 
will  have  named  the  number  of  sticks  we  gave  him. 

For  example,  if  he  has  made  three  bundles,  and  eight 
sticks  are  left  over,  he  will  say,  looking  to  the  left, 
"  thirty  "  ;  looking  to  the  right,  "  eight  "  ;  then,  without 
a  stop,  "  thirty-eight." 

Having  repeated  this  exercise  very  often,  with  groups 
of  sticks  taken  at  chance,  we  will  take  a  faggot  to  pieces, 
and  propose  to  count  the  sticks  successively  one  by  one. 
We  will  begin  by  counting  one,  two,  three,  ...  to 
ten.  Then,  having  a  bundle,  we  will  place  it  on  the  left 
(without  even  needing  to  tie  it)  and  go  on,  saying  : — 

One-ten  ;  two-ten l ;  thirteen  ;  fourteen  ;  fifteen; 
Sixteen;  seventeen;  eighteen;  nineteen. 

At  last  a  fresh  stick  finishes  a  second  bundle,  which 
we  put  on  the  left,  by  the  side  of  the  first,  saying  twenty  ; 
we  go  on  in  the  same  way  to  the  ninth  bundle,  then  to 
the  ninth  stick  left,  which  we  touch  saying  ninety-nine  ; 
finally  we  remove  the  last,  finishing  the  tenth  bundle, 
which  we  put  on  the  left,  by  the  side  of  the  nine,  saying 
the  word  hundred. 

There  is  nothing  to  prevent  us  telling  the  young  pupil 
then  that  we  have  just  taught  him  numeration  from  one 

1  Here  we  must  not  say  "  eleven,  twelve."  These  names  can  be  learnt 
without  any  trouble  at  the  right  moment.  It  is  useless  to  load  the 
memory  with  them  now.  Even  if  the  child  in  his  logic  says  "  three- 
ten,  four-ten,"  etc.,  there  is  no  need  to  correct  him  yet. 


8  MATHEMATICS 

to  a  hundred ;  we  can  even  tell  him  that  when  he  says 
"  seventy-three  "  matches  or  sticks,  he  is  performing  spoken 
numeration,  and  that,  when  he  puts  seven  bundles  to  the 
left  and  three  sticks  to  the  right,  he  is  performing 
numeration  by  figures.  He  will  be  all  the  more  flattered 
to  feel  himself  so  wise  since  he  does  not  yet  know  how  to 
write  a  letter  or  a  figure,  or  to  read,  b,  a,  ba.  But  he 
draws  strokes,  he  has  eyes,  he  uses  them  to  see,  and 
begins  to  understand  what  he  sees  and  what  he  is  doing. 

So  now  we  know  how  to  count  sticks  from  one  to  a 
hundred.  We  must  accustom  ourselves  to  count  any 
other  things  in  the  same  way,  and  then  to  count  them  in 
our  heads  at  once  without  having  them  before  our  eyes. 
That  is  the  beginning  of  mental  calculation,  so  important 
in  practice  and  so  easy  to  make  children  practise  from 
the  earliest  age,  if  we  begin  with  very  simple  things  and 
go  on  gradually. 

This  is  not  yet  all ;  starting  from  one,  we  must  become 
accustomed  to  count  by  twos  : — 

One,  three,  ...  to  ninety-nine, 

and  explain  that  all  these  numbers  are  odd  numbers. 
We  must  do  the  same,  starting  from  two  : — 

Two,  four,  six,  ...  to  a  hundred, 

and  we  will  have  the  even  numbers.  We  will  then  become 
accustomed  to  count  by  threes,  by  fours,  starting  at  first 
from  one,  and  then  from  any  number. 

All  these  exercises  are  to  be  done  first  with  things — 
preferably  sticks — then  mentally. 

In  short,  this  manipulation  of  numbers,  from  one  to  a 
hundred,  can  be  varied  indefinitely,  for  we  must  not  fear 
prolonging  it  as  long  as  it  does  not  become  tedious  and 
as  Jong  as  it  interests  the  child.  It  will  be  well  to  come 
back  to  it  from  time  to  time,  even  when  he  has  penetrated 
a  little  further  in  his  introduction  to  science. 


THE  ADDITION  TABLE 


III  ,>::!; ! 


I  IM 


jIH 


HI! 


5.  The  Addition  Table. 

Let  us  arrange  on  a  table,  from  left  to  right,  one, 
two,  ...  to  nine  sticks,  separating  these  nine  groups 
from  one  another.  Below  the  single  stick,  let  us  put  two 
and  make  a  column,  starting  two,  three,  to  ten.  A  second 
column  made  in  the  same  way  will  contain  three,  four,  .  .  . 
one-ten  sticks ;  and  going  on  in  the  same  way  we  will 
have  nine  columns ;  the  last  group  of  the  ninth  column 
will  be  of  eighteen  sticks. 

Now  is  the  moment  to  come  back  and  use  our  skill  in 
drawing  and  our  great  aptitude  in  tracing  strokes.  Only, 
as  it  is  troublesome  to  draw 
the  ten  strokes  representing 
the  sticks  in  a  bundle,  we 
will  make  a  picture  of  a 
bundle  by  a  thick  stroke, 
stronger  than  the  others, 
made  of  two  parts,  H,  with 
a  little  bar  to  recall  the 
presence  of  the  rubber  band. 
Thus  we  have  now  begun 
to  know  how  to  write  num- 
bers with  strokes,  and 
in  this  way  copying  the 
figure  as  we  are  going  to 
explain,  we  will  have  Fig.  5,  at  least  partly.  To  finish 
it,  we  will  put  one,  two,  .  .  .  nine  strokes  on  the  left 
of  the  two,  three,  .  .  .  ten  of  the  first  column;  finally, 
we  will  separate  this  new  column  by  a  vertical  line  from 
the  rest  of  the  figure,  and  we  will  also  separate  the  first 
row  by  a  horizontal  line. 

The  figure  we  have  thus  obtained  is  an  addition  table  ; 
we  will  soon  see  why  it  is  so  called. 

It  lends  itself  to  several  interesting  remarks  which 
the  maker  will  partly  find  out.  First,  all  the  numbers 
in  the  same  slanting  line,  rising  from  left  to  right,  are  the 


Kin 


FIG.  5. 


10  MATHEMATICS 

same ;  further,  all  the  numbers,  read  from  left  to  right 
in  a  horizontal  line,  or  from  top  to  bottom  in  a  column, 
are  the  numbers  counted  one  by  one  ;  finally,  the  numbers 
in  the  same  slanting  line  going  down  from  left  to  right,  if 
read  downwards,  are  the  numbers  counted  two  by  two. 
These  are  sometimes  even,  sometimes  odd. 

Nothing  stops  us  from  reading  all  these  numbers  in 
the  reverse  order,  which  will  teach  us  to  say  the  numbers 
fluently,  one  by  one,  or  two  by  two,  in  the  direction 
opposite  to  the  natural  order.  There  is  still  a  very 
important  exercise  of  which  we  have  not  yet  spoken,  and 
which  we  will  now  begin  with  small  numbers  ;  this  will  not 
present  any  serious  difficulty. 

6.  Sums. 

Let  us  take  two  heaps  of  beans  (or  other  things)  and 
count  them  both.  If  we  put  them  all  into  one  heap, 
how  many  shall  we  have  ?  For  this,  we  have  only  to  count 
again  the  heap  made  by  mixing  the  other  two.  But 
this  would  be  lengthy  and  wearisome,  and  it  would  be 
time  lost. 

We  will  explain  that  there  is  a  quicker  way  of  getting 
the  result,  that  we  will  do  it  by  an  operation  called 
addition,  and  that  the  number  of  things  in  the  big  heap, 
which  we  want  to  find,  is  called  the  sum  or  total. 

Taking  numbers  smaller  than  ten,  and  looking  again 
at  Fig.  5,  we  notice  that  it  gives  all  the  sums  of  two  heaps, 
and  we  will  invite  the  child  to  try  and  find  this  out.  We 
will  do  this,  repeating  these  exercises  as  often  as  possible 
and  making  him  count  the  sum  itself  when  he  does  not 
find  it. 

Even  before  this  addition  table  is  completely  fixed  in 
the  memory,  we  will  take  any  two  numbers — chosen  so 
that  their  sum  is  less  than  a  hundred — and  we  will  count 
them  both  separately.  We  will  then  represent  them  by 
sticks ;  suppose  they  are  thirty-four  and  twenty-three. 


SUMS  11 

The  first  number  is  made  up  of  three  bundles  and  four 
sticks ;  the  other,  of  two  bundles  and  three  sticks,  is 
placed  below — bundles  under  bundles,  on  the  left,  sticks 
under  sticks,  on  the  right. 

We  will  then  ask  the  child  to  say  how  many  are  made  by 
four  and  three  sticks  ;  he  will  answer  "  seven,"  helping  him- 
self if  necessary  by  the  addition  table,  and  he  will  place 
seven  sticks  a  little  lower  down.  So,  how  many  do  three 
and  two  bundles  make  ?  Five  bundles,  which  we  will 
place  below  the  bundles.  We  have  thus  the  total,  five 
bundles,  seven  sticks,  or  fifty-seven  sticks. 

We  will  begin  again  with  other  numbers,  taking  those 
where  there  are  only  bundles  and  no  sticks,  like  sixty, 
twenty,  eighty ;  others  where  there  are  no  bundles, 
numbers  less  than  ten ;  but  so  that  each  sum  of  bundles 
or  sticks  is  always  less  than  ten. 

When  we  have  reached  this  point,  we  will  take  other 
numbers  where  this  is  not  so ;  for  instance,  forty-nine 
and  twenty-five. 

The  operation  is  performed  thus  : — 

Four  bundles  Nine  sticks 

Two  bundles  Five  sticks 

We  have  then  nine  and  five,  or  fourteen  sticks ;  this 
gives  us  a  bundle — which  we  put  under  the  bundles — and 
four  sticks.  Then  counting  the  bundles,  beginning  with 
that  we  have  just  made,  we  have  one  and  four,  five ; 
five  and  two  bundles,  seven.  The  total  is  thus  seven 
bundles  and  four  sticks,  or  seventy-four. 

This  exercise  should  be  repeated,  renewed  with  different 
examples,  so  that  it  will  interest  the  child  without  boring 
him. 

Then,  coming  to  additions  of  several  numbers,  we  will 
proceed  in  the  same  way  (always  arranging  that  the  total 
is  less  than  a  hundred),  and  we  will  see  that  thus  we  find 


12  MATHEMATICS 

the  number  formed  by  joining  several  heaps,  when  we 
know  the  number  in  each  heap.1 

Repeat  these  exercises  on  a  crowd  of  examples  as  long 
as  they  do  not  cause  fatigue  or  boredom.  If  the  child 
seerns  to  be  at  all  troublesome,  the  punishment  should 
consist  in  a  threat — carried  out  for  several  days — not  to 
go  gn  showing  him  the  game  of  sticks,  counters,  etc., 
which  he  has  begun  to  learn.  Let  this  device  be  used 
with  some  skill  and  we  will  see  that  it  is  not  hard  to  lead 
the  offenders  back  to  their  studies  by  their  own  wishes^ 
Only,  do  not  pronounce  in  their  ears  the  unfortunate 
word  "  study,"  which  might  frighten  them. 


7.  Subtraction. 

• 

I  have  a  big  heap  of  counters,  eighty-seven  let  us  say  ; 
I  pick  up  or  take  away  a  few  which  I  count :  I  find  there 
are  twenty-five.  How  many  are  left  ?  To  find  that  is  to 
do  a  subtraction  ;  the  result  is  the  remainder  or  difference ; 
we  notice  that  if  we  put  back  the  remainder  to  the  number 
from  which  we  have  taken  it  we  make  once  more  the 
big  heap,  by  which  I  mean  the  number  from  which  I 
subtract.  To  find  the  difference,  first  let  us  write  the 
larger  number,  eighty-seven,  with  little  sticks : — 

Eight  bundles  Seven  little  sticks, 

and,  underneath,  the  smaller  one,  twenty-five  : — 
Two    bundles  Five    little    sticks, 

taking  great  care  to  put  the  bundles  to  the  left,  the  little 
sticks  to  the  right,  and  to  put  the  little  sticks  under  each 
other  and  the  bundles  under  each  other.  From  the 
larger  number  I  take  away  five  little  sticks ;  I  shall  have 

1  These  exercises  will  oblige  the  child  to  learn,  beyond  the  addition 
table,  how  to  add  quickly  a  number  less  than  ten  to  a  number  less  than 
a  hundred  ;  for  instance,  sixty-eight  and  five,  seventy-three.  With 
a  little  patience  this  result  will  be  obtained  by  practice  quickly  enough. 


SUBTRACTION  13 

two  left.  I  take  away  two  bundles,  and  six  will  remain. 
Then  the  remainder  will  be 

Six  bundles        Two  little  sticks, 

or  sixty-two  little  sticks. 

Now  we  are  able  to  subtract  (only  doing  it  with 
numbers  less  than  ten),  since  we  have  taken  away  five 
from  seven,  and  two  from  eight.  But  it  isn't  always  so 
easy  !  For  instance,  the  big  heap  may  count  up  to  fifty- 
two,  and  what  we  wish  to  take  away  may  be  eighteen, 
which  is  certainly  less.  Arranging  them  as.  before  we 
have 

Five    bundles         Two    little    sticks ; 

One    bundle           Eight    little    sticks. 

We  cannot  take  away  eight  sticks  from  two.  So  from 
among  the  five  bundles  we  take  one  which  we  put  to  the 
right  with  the  two  sticks.  Whether  we  undo  it  or  not 
we  can  see  very  well  that  now  we  have  ten-two  sticks  to 
the  right,  and  only  four  bundles  to  the  left  instead  of 
five.  Now  from  the  ten-two  little  sticks  at  the  right  we 
shall  take  away  eight.  We  shall  have  four  left  j  from 
the  four  bundles  at  the  left  we  take  away  one  :  there 
are  three  left.  The  remainder  then  is 

Three   bundles         Four   little   sticks, 

or  thirty-four. 

We  must  know  how  to  subtract  a  number  less  than  ten 
from  a  number  larger  than  ten,  but  which  will  be  always 
less  than  twenty.  By  multiplying  these  exercises  many 
times,  and  by  varying  them  as  much  as  possible,  this  sub- 
traction (which  has  to  be  learnt)  will  be  remembered  by  the 
pupil ;  but  above  all,  do  not  cause  them  to  be  learnt  by 
heart  and  recited.  Do  them  all  the  time  instead  :  this 
is  much  more  effective.  We  must  take  care  always  to 
take  for  the  larger  number  one  less  than  a  hundred, 
because  at  present  we  cannot  count  beyond  that. 


14  MATHEMATICS 

8.  Thousands  and  Millions. 

Up  to  now  we  know  how  to  count  up  to  one  hundred.  It 
is  a  big  number  if  we  consider  the  age  of  a  person  in  years  ; 
a  man  aged  one  hundred  is  very  old,  and  centenarians  are 
very  rare.  But  it  is  a  very  little  number  if  we  count  grains 
of  corn ;  a  heap  of  a  hundred  grains  of  corn  is  not  at  all 
big,  and  would  not  be  sufficient  to  feed  a  child  for  a  day, 
so  that  it  is  impossible  to  stop  there,  and  we  shall  be  obliged 
to  climb  still  further  up  the  ladder,  which  will  not,  however, 
be  difficult. 

We  have  got  up  to  one  hundred  by  grouping  the  little 
sticks  in  bundles  of  ten,  and  grouping  ten  bundles  of  those 
in  a  faggot,  which  contains  one  hundred  little  sticks. 
Let  us  put  together  ten  faggots  in  a  box,  then  with  ten  such 
boxes  let  us  form  a  bale,  ten  bales  can  be  put  together  in 
a  basket,  with  ten  baskets  we  will  make  a  case,  with  ten 
such  cases  we  will  make  a  wagon-load,  with  ten  wagon- 
loads  a  car-load,  and  with  ten  car-loads  a  train. 

Going  all  over  this,  we  are  going  to  give  the  names  of 
the  numbers  that  we  obtain  in  this  manner. 

A  match  or  a  stick  is  what  we  will  call  a  simple  unit. 

In  a  bundle  we  have  ten  matches,  or  one  set  of  ten. 

In  a  faggot  of  ten  bundles,  one  hundred  matches,  or  a 
set  of  one  hundred. 

In  a  box  of  ten  faggots,  a  thousand  matches. 

In  a  bale  of  ten  boxes,  ten  thousand  matches  or  one  ten 
of  thousands. 

In  a  basket  of  ten  bales,  one  hundred  thousand  or  one 
hundred  of  thousands. 

In  a  case  of  ten  baskets,  a  million. 

In  a  wagon-load  of  ten  cases,  ten  millions,  or  one  ten 
of  millions. 

In  a  car-load  of  ten  wagon-loads,  one  hundred  millions, 
or  a  hundred  of  millions. 

In  a  train  of  ten  car-loads,  one  thousand  millions. 

We  might  go  on  as  long  as  we  liked,  but  the  number — 


THOUSANDS  AND   MILLIONS  15 

a  thousand  millions — at  which  we  have  arrived,  is  large 
enough  for  ordinary  use.  We  should  have  an  idea  of  its 
size  if  we  placed  ordinary  wooden  matches,  one  after 
another,  to  the  number  of  one  thousand  millions ;  the  total 
length  would  be  considerably  more  than  the  circumference 
of  the  earth. 

Trying  to  count  a  thousand  million  matches  one  by  one, 
supposing  that  we  took  a  second  for  each,  and  occupying  in 
this  counting  ten  hours  a  day,  we  would  take  more  than 
seventy-six  years.  This  would  perhaps  be  rather  long, 
not  very  amusing,  and  only  slightly  instructive. 

No,  if  we  want  to  count  a  big  heap  of  little  sticks,  we 
will  make  bundles  of  them,  and  we  will  put  to  the  right 
the  little  sticks  that  are  left  over ;  once  the  bundles  are 
made,  perhaps  there  may  be  three  little  sticks.  We  will 
now  make  faggots  with  our  bundles,  making  them  up  in 
tens ;  suppose  there  remain  eight  bundles,  we  will  place 
them  to  the  left  of  the  three  little  sticks  ;  we  will  count 
our  faggots  in  tens  to  make  boxes  of  them.  Five  faggots 
are  left,  we  place  them  to  the  left  of  the  eight  bundles 
and  counting  our  boxes  we  find  six  of  them.  We  put 
them  to  the  left  of  the  five  faggots  and  we  have  thus 
the  number  of  little  sticks :  six  boxes,  five  faggots, 
eight  bundles,  three  little  sticks  ;  or  six  thousand  five 
hundred  and  eighty-three  little  sticks.  With  nothing 
but  bundles  and  faggots  we  shall  be  able  to  count  up  to  a 
thousand,  and  form  all  the  numbers  up  to  that,  never  for- 
getting that  faggot,  bundle,  single  little  stick  mean 
respectively 

a    hundred,    ten,    one,    little    sticks. 

If  in  the  number  that  we  wish  to  unite  there  are  no 
single  little  sticks,  or  no  bundles,  that  will  not  make  any 
difference.  For  instance, 

Eight  faggots,  six  bundles, 

Will  contain  eight  hundred  and  sixty  little  sticks,  and 

Five  faggots,  three  little  sticks 

Will  contain  five  hundred  and  three. 


16  MATHEMATICS 

It  will  be  necessary  to  have  numbers  under  a  thousand 
formed  like  this,  and  to  have  performed  many  additions 
and  subtractions,  exactly  as  it  has  been  shown  before,  but 
extending  the  method  of  procedure  as  far  as  faggots 
instead  of  keeping  to  bundles. 

It  is  advisable  to  notice  that  we  often  meet  the  same 
numbers,  ten  and  hundred,  or  tens  and  hundreds.  Thus : 

Little  stick  |  (  one 

Bundle  I  mean     -j  one  ten 

Faggot  j  I  one,  hundred 

Box  |  f  one  thousand 

Bale  „        •!  one  ten  of  thousands 

Basket  {  one  hundred  of  thousands 

Case  1  f  one  million 

Wagon-load  „         -j  one  ten  of  millions 

Car-load  [  one  hundred  of  millions 

A  number  of  thousands  or  of  millions  will  be  reckoned 
then  as  if  we  were  counting  simple  little  sticks  from  one 
to  a  thousand.  Thus  : — 

Three  car-loads     Two  wagon-loads     Seven  cases 
One  basket          (no  bales)  Nine  boxes 

Four  faggots      Five  bundles 

will  be  a  number  of  little  sticks  which  will  express 

Three  hundred  and  twenty-seven  millions"!  ,.    , 
One  hundred  and  nine  thousand  .  , 

Four  hundred  and  fifty 

We  could  have  some  of  them  counted  like  that,  but 
without  insisting  upon  the  large  numbers  for  the  moment, 
and  applying  ourselves  particularly  to  the  bundles,  and 
faggots,  going  no  further  than  the  boxes  at  the  very  most. 

Always,  in  what  has  gone  before,  we  have  taken  care 
to  place  the  little  sticks — single  ones — to  the  right,  the 
bundles — tens — to  the  left ;  the  faggots — hundreds — to 
the  left  of  the  bundles,  and  so  on.  Strictly  speaking,  we 


COLOURED  COUNTERS  17 

ought  to  notice  that  this  is  unnecessary,  but  it  is  most 
convenient,  and  it  is  a  good  thing  to  keep  to  this  arrange- 
ment always,  because  the  calculation  is  done  in  order. 

A  little  later,  the  child,  having  been  accustomed  to  this 
habit,  will  find  it  natural ;  this  will  be  valuable,  as  it  will 
be  indispensable  when  calculating.  To  form  effectively, 
with  little  sticks,  all  the  numbers  of  which  we  have  just 
spoken,  and  of  which  it  is  a  good  thing  to  speak  to  settle 
the  child's  mind,  we  must  find  something  rather  heavy, 
and  very  easy  to  place  on  a  table,  or  on  a  sheet  of 
paper,  even  before  making  it  up  into  car-loads.  We  are 
going  to  see  now  how  we  can  simplify  things,  and  show 
the  young  mathematician — who  cannot  either  read  or 
write  fluently — that  it  is  perfectly  easy  to  manage  with 
his  fingers  the  enormous  numbers  with  which  he  has  to 
deal. 

9.  Coloured  Counters. 

It  is  very  disagreeable  to  be  so  encumbered,  as  soon 
as  we  want  to  count  a  thousand  matches,  by  our  bundles 
and  faggots.  As  we  know  already  that  numbers  can 
be  represented  by  any  means,  let  us  replace  our  matches 
by  white  counters.  That  does  not  alter  our  calculations 
nor  the  manner  of  doing  them.  Now,  let  us  change  our 
bundles  for  red  counters ;  they  will  be  really  more  con- 
venient to  manage,  and  we  can  always  replace  a  red  by  ten 
white  ones  if  it  is  necessary.  To  carry  it  still  further, 
instead  of  faggots  we  will  put  orange  counters  ;  instead 
of  boxes,  yellow  counters  ;  instead  of  bales,  green  counters; 
instead  of  baskets,  blue  counters ;  instead  of  cases, 
indigo  counters  ;  instead  of  wagon-loads,  violet  counters ; 
instead  of  car-loads,  black  counters ;  and  finally,  instead 
of  trains,  long  counters — white  ones. 

The  objects  and  the  numbers  correspond  in  this  manner  : 

(Trains,  car-loads,  wagon-loads,  cases, 
baskets,  bales,  boxes,  faggots,  bundles 
and  matches. 

M.  C 


18 


MATHEMATICS 


Counters 


Numbers 


jLong,  black,  violet,  indigo,  blue,  green, 
(     yellow,  orange,  red,  white. 

f  Thousands    of    millions,    hundreds    of 

millions,    tens    of    millions,    millions, 

j      hundreds  of  thousands,  tens  of  thou- 

(•     sands,  thousands,  hundreds,  tens,  units. 


Nothing  hinders  us  then  from  writing  all  the  numbers  that 
we  require  up  to  a  thousand  millions,  and  even  further 
than  that,  with  our  little  counters,  without  being  compelled 
to  use  cases,  car-loads,  and  even  trains  ;  and  it  is  equally 
in  our  power,  if  that  will  interest  us,  to  add  and  subtract. 
It  will  be  necessary,  though,  always  to  remember  that  a 
red  counter  is  equal  to  ten  white,  an  orange  counter  to 
ten  red,  and  so  on  to  the  end. 

It  might  seem  that  for  white  counters,  we  might 
put  cents,  then  replace  the  red  counters  by  dimes, 
and  continue  like  that ;  but  that  would  become  awkward 
and  cumbersome,  and  we  would  be  obliged  to  have 
a  nice  little  fortune  ;  because  then,  to  represent 
thousands  of  millions,  we  would  have  to  use  coins 
each  worth  ten  million  dollars.  Money  of  that  value 
is  not  coined,  it  would  be  difficult  to  handle,  and 
it  will  be  decidedly  better  to  continue  to  use  long  white 
counters  to  represent  thousands  of  millions.  It  will  also 
be  more  economical !  Always,  as  we  go  higher,  we  will 
put  our  counters  carefully  in  order,  beginning  at  the 
right. 


be 

a 

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h5 

^j 

o 
o3 

M 

-1-3 

jy 
"o 

> 

Indigo. 

<u 

3 

5 

d 

<D 
g 

O 

Yellow. 

<u 

1 

2 

O 

H3 

u 

tf 

1 

£ 

FIGURES  19 

Simply  by  looking  at  each  place  we  know  what  colour 
ought  to  go  there  according  to  its  place,  starting  from 
the  right. 

10.  Figures. 

We  know  now  how  to  write  all  the  numbers,  at  least 
as  far  as  thousands  of  millions — and  it  would  be  easy  to 
go  on  much  further — with  our  round  counters  of  various 
colours,  and  the  long  white  counters.  To  do  that,  we 
must  put  in  each  of  the  places  where  we  see  the  white 
or  red  counters,  etc.,  or  units,  and  tens,  etc.,  a  number  of 
counters  which  is  always  less  than  ten. 

If  by  any  means  we  could  avoid  reckoning  these 
counters  every  time,  it  would  be  much  more  convenient. 
By  this  time  the  pupil  has  commenced  to  write  a  little, 
and  we  can  exercise  him  in  tracing  the  characters  which 
will  represent  the  first  nine  numbers,  which  we  shall 
need — characters  which  are  called  "figures."  These  are  : — • 

One    two    three    four    five    six    seven    eight    nine 
1234567  8  9 

Whether  with  a  pencil  or  with  a  pen,  he  will  become 
accustomed  to  form  them  correctly  without  any  flourish- 
ing, with  a  single  stroke,  except  perhaps  the  figures  four 
and  five,  which  require  two,  making  use  of  ruled  slates 
or  ruled  paper  at  first,  so  that  the  figures  may  be  all 
the  same  height.  This  is  of  the  highest  importance  for 
the  future  practice  of  calculations.  Here  is  the  type  to 
which  we  must  keep  : — 

12,34-5(3789 

Just  for  curiosity  we  may  notice  here  that  all  the 
figures,  according  to  some  old  authors,  owe  their  origin 

c2 


20  MATHEMATICS 

to  the    figure       ^     (see  below) ;    but  this  is  not  well 
authenticated. 


12)345678        9 

The  important  point  is  to  make  the  pupil  change  the 
numbers  from  little  sticks  into  counters,  and  from  counters 
into  figures,  not  taking  very  large  numbers,  especially  to 
begin  with.  We  will  notice  that  there  is  no  necessity 
to  make  our  figures  of  different  colours,  because  the  place 
which  they  occupy  tells  us  quite  easily  whether  they 
represent  simple  units,  tens,  hundreds,  etc.,  or  counters 
coloured  white,  red,  orange,  etc.,  or,  again,  little  sticks, 
bundles,  faggots,  etc. 

But  now  comes  an  important  observation.  If  there  are 
no  counters  at  all  of  a  certain  colour,  we  don't  put  any- 
thing there.  There  is  only  the  place  in  the  row  to  distin- 
guish the  figures ;  if  we  put  nothing  there  it  would  mix 
everything  up,  because  we  ought  to  leave  a  space  always 
the  same  size,  that  of  a  figure,  and  we  are  not  clever 
enough  to  write  always  so  regularly.  Besides,  if  the 
space  happened  to  be  in  the  unit  place,  how  could  we 
know  the  meaning  of  the  last  figure  to  the  right  ?  To 
escape  all  these  difficulties,  we  put,  in  empty  spaces,  a 
round  character,  0,  that  we  call,  "  zero  '51  which  has  no 
value  but  fills  up  the  place.  Zero  is  a  good  modest 
servant  who  guards  the  house,  and  who  says  to  you  : 
"  There  is  nobody  here.  As  for  me,  I  do  not  count, 
I  am  nothing,  but  I  hinder  anybody  from  coming 
in." 

From  now  onward  we  can  teach  the  pupil  to  write 
quantities  of  numbers  often  using  zero  or  "  nought," 
and  varying  exercises  of  this  kind.  If  there  are  several 
pupils,  one  might  exercise  them  together,  rousing 

1  The  inventor  of  the  zero  is  not  known,  but  this  clever  idea  seems  to 
be  of  Hindoo  origin. 


FIGURES  21 

their  emulation  a  little,  leading  them  on,  more  and 
more,  to  read  and  write  quickly  and  correctly,  and 
tell  them  at  the  end  of  the  calculation  that  they  now 
understand  written  numeration. 

Having  arrived  at  this  stage,  it  is  a  good  thing  to  go 
back  again  to  the  examples  of  addition  and  subtrac- 
tion that  we  have  been  able  to  express  with  little  sticks, 
or  counters,  making  use  now  of  figures.  But  there  will 
be  various  very  useful  observations  to  make  which  formerly 
would  not  have  been  of  any  service.  One  of  them,  in 
doing  addition,  consists  in  accustoming  the  pupil  to  speak 
as  little  as  possible,  never  to  say,  for  instance,  "I  put 
down  such  and  such  a  figure,  and  I  carry  such  and  such  a 
number."  It  will  be  sufficient  to  make  oneself  understood 
to  give  the  example  of  addition  shown  here  : 

3087 
6944 

560 

208 

29 

2004 


12832 

Which  ought  to  be  translated  then  in  spoken  language  : 
"  The  figures  7  and  4  =  one-ten,  and  8  =  nine-ten,  and 
9  =  twenty-eight,  and  4  =  thirty-two "  (we  write  2 
without  saying  anything) ;  then  we  add,  "  I  carry  3,  and 
8  =  one-ten,  and  4  =  five-ten,  and  6  =  twenty-one,  and 
2  =  twenty-three"  (we  write  3);  "I  carry  2,  and  9  = 
one-ten,  and  5  =  six-ten,  and  2  =  eight-ten  "  (we  write 
8) ;  "I carry  1,  and  3  =  4,  and 6  =  ten,  and  2  =  two-ten  " 
(we  write  2,  then  1  to  the  left  of  that);  and  we  read 
the  total,  "two-ten  thousand,  eight  hundred  and 
thirty-two." 

A     second     remark     applies     to     the     practice    of 


22  MATHEMATICS 

subtraction,  when  there  is  in  the  greater  number,  in  a 
certain  row,  a  figure  less  than  the  one  below.  Let  us 
go  back  to  the  example  in  section  7 ;  from  52  we  have  to 
take  18. 

52  4  12 

18  1  8 

34  3  4 

What  we  have  done  with  our  little  sticks  is  shown 
above.  But  we  must  not  write  anything  else  than  52  and  18 
before  the  result  of  the  operation,  and  it  might  very  easily 
happen  that  we  forget  that  we  have  taken  away  a  ten  from 
the  top,  and  there  only  remain  four  tens  instead  of  five. 
For  the  future  we  proceed  in  another  way,  noticing  that 
to  take  1  from  4  is  the  same  thing  as  taking  2  from  5. 
And  we  would  say,  "8,  from  two-ten,  equals  4.  I  carry 
1,  1  and  1  make  2,  2  from  5  equals  3."  Then  instead  of 
saying  "1  from  4,"  I  say  "2  from  5,"  which  leaves  3; 
so  we  get  into  the  habit  of  carrying  one  each  time  that  we 
have  previously  added  a  10  to  the  figure  from  above. 

Many  exercises  in  addition  and  subtraction  ought  to 
be  carried  out  like  this.  The  child  will  interest  himself  in 
them,  but  do  not  try  to  prove  anything  to  him.  If  he 
seems  at  times  puzzled,  take  him  back  to  his  sticks  or  to  his 
counters  ;  and  try  only  to  give  him  practice  in  calculation 
and  not  to  make  him  learn  words  that  he  does  not  under- 
stand. If  any  observations  come  into  his  mind,  and  he 
tells  you  of  them,  listen  to  him  with  great  attention. 
Do  not  be  afraid  of  going  back  from  time  to  time,  in  order 
to  accustom  him  to  compare  his  numbers  written  in 
figures  with  collections  of  little  sticks,  counters,  or  any 
other  objects.  And,  above  all,  do  not  make  the  lesson 
too  long ;  do  not  let  his  interest  flag  or  fatigue  to  overcome 
him ;  this  is  the  teacher's  deadliest  scourge. 

If  you  think  it  convenient,  you  can,  from  this  time, 


STICKS   END  TO  END  23 

though  there  is  no  hurry  about  it,  initiate  the  pupil  into 
the  ordinary  names  for  the  numbers  11  and  12. 


11.  Sticks  End  to  End. 

Let  us  make  use  once  more  of  the  little  sticks  that  we 
have  already  employed  ;  we  will  imagine  that  we  have, 
say,  three  heaps  in  which  there  are  5,  3  and  4  little 
sticks.  If  we  put  all  the  little  sticks  one  after  another  in 
the  same  direction,  the  length  of  this  row  will  be  twelve 
little  sticks,  that  is  to  say,  it  will  give  the  sum  of  the 
numbers  represented  by  the  three  heaps. 

We  should  arrive  at  the  same  result  if  we  replaced  the 
little  sticks  belonging  to  the  first  heap  by  a  straw  equal 
in  length  to  the  five  little  sticks  ;  those  from  the  second 
heap  by  one  as  long  as  the  three  little  sticks ;  and  those 
from  the  third  heap  by  a  straw  measuring  the  length  of 
the  four  little  sticks. 

If,  instead  of  these  very  little  numbers  we  took  larger 
ones,  and  if,  instead  of  three  numbers  we  took  as  many  as 
we  liked,  all  that  we  have  just  said  would  repeat  itself. 
The  straws  would  be  longer;  there  would  be  more  than 
three  straws ;  that  is  all  the  difference. 

We  prove  in  this  manner  that  any  number  whatever  can 
be  represented  by  a  straw  of  suitable  length,  and  that  to 
find  the  sum  of  several  numbers  we  have  nothing  to  do 
but  lay  end  to  end,  one  after  the  other,  the  straws  which 
represent  these  numbers.  The  length  of  the  line  of 
straws  thus  obtained  will  be  the  desired  sum. 


12.  The  Straight  Line. 

The  straws  of  which  we  have  just  been  speaking  in 
the  foregoing  operations  ought  always  to  be  placed  in  a 
straight  line  immediately  after  each  other.  But  what  is 
a  straight  line  ?  We  have  an  idea  of  it  by  the  stroke 


24  MATHEMATICS 

which  a  very  fine-pointed  pencil  makes  moving  along 
against  a  ruler  held  horizontally,  or  by  an  extremely 
fine  thread,  a  hair,  for  instance,  held  between  two  supports. 
This  general  idea  is  sufficient  for  us  ;  we  know  quite 
well  that  if  the  ruler  were  longer,  the  sheet  of  paper 
wider,  we  would  be  able  to  draw  our  straight  line  further, 
either  to  one  end  or  to  the  other  :  and  as  there  is  no  need 

ever    to    stop,  we    under- 

-         -^ stand     that     the     straight 

line    is,     so    to    speak,    an 

,-      c  indefinite   figure.     We   will 

*IG-6-  i  t    -L 

never  make    use  of    it  any 

further  than  the  limit  that  we  require ;  but  this  limit 
can  be  as  distant  as  we  wish. 

If  we  take  a  straight  line  (Fig.  6)  and  mark  off  a  point 
A,  and  another  point  B,  the  portion  of  the  straight  line 
AB  comprised  between  these  two  points  is  what  we  call 
a  segment  of  a  straigh  line.  The  straws  which  we  made 
use  of  just  now  can  be  laid  on  the  segments  of  the  straight 
line,  and  the  length  of  these  straws  is  the  same  as  the 
segments  upon  which  they  are  laid. 

Thus  (Fig.  7),  to  return  to  the  example  (section  11),  let 
us  take  a  straight  line  upon  which  we  mark  a  point  O, 
no  matter  where  :  starting  from  this  point,  let  us  take 
a  segment  OA,  which  is  of 

the  same  length  as  our  first  o  '   '  '"'  A  '"""a"  '"*''" Q 

straw,    five    little    sticks  ; __ 

starting  from  A,  let  us  take  'f  '_      ;._ 

a  segment   AB,  having   its  „      _ 

length  the  same  as  that  of 

the  second  straw,  three  little  sticks ;    then  starting  from  B, 

another,  BC,  of  which  the  length  equals  that  of  the  third 

straw,  four  little  sticks.     The  segment  OC  will  be  the 

length  of  twelve  little  sticks ;    the  sum  of  5,  3  and  4. 

Whether  we  say  that  we  add  numbers,  straws,  segments  of 

a  straight  line,  it  is  always  the  same  thing,  the  addition 

is  made  by  laying  the  straws,  or  the  segments  end  to  end, 


SUBTRACTION  WITH  LITTLE  STICKS       25 

one  after  another.  This  operation  must  necessarily  be 
done,  if  with  segments,  always  in  the  same  direction ;  we 
will  suppose  it  be  invariably  from  left  to  right. 

In  Fig.  7  we  can  thus  go  on  adding  as  far  as  we  like 
to  the  right  of  O,  but  never  to  the  left. 


13.  Subtraction  with  Little  Sticks. 

It  is  not  any  more  difficult  to  subtract  than  to  add, 
by  the  aid  of  little  sticks.  Suppose,  for  example,  that  we 
want  to  take  4  from  11.  We  will  lay  11  sticks  end  to 
end  in  a  straight  line ;  then,  beginning  at  the  end  to  the 
right  of  this  row,  we  take  away  4  little  sticks ;  a  row  of 
7  little  sticks  remains ;  7  is  the  difference  between  11 
and  4. 

If  we  begin  by  putting  a  straw  as  long  as  11  little  sticks, 
it  would  seem  as  though  we  were  obliged  to  cut  off  an 
end  of  it  equal  in  length  to  the 
four  to  make  the  difference.  ,  .         .       a 

But   there    is    another    way          °  * 

which  we  will  understand  at  FIG.  8. 

once,  using  segments  instead 

of  straws.  Let  us  lay  out  on  a  straight  line,  beginning 
from  the  point  O,  a  segment  OB  the  length  of  11  little 
sticks.  Starting  from  B,  let  us  take  a  segment  the  length 
of  4,  but  instead  of  supposing  it  traced  from  left  to  right, 
let  us  take  it,  on  the  contrary,  from  right  to  left.  The 
segment  OC  will  represent  by  its  length  the  difference  7. 

We  go  over  this  a  few  times,  saying  that  to  add  several 
segments  we  must  lay  them  end  to  end  in  the  same 
direction ;  and  to  subtract  one  segment  from  another 
we  must  lay  them  end  to  end  in  the  same  manner,  but  in 
the  opposite  direction. 

These  things,  besides  being  easy,  are  perfectly  under- 
standable ;  it  suffices  to  vary  the  examples  a  little  from 
time  to  time  to  interest  the  pupil ;  we  must  not  be  afraid 


26  MATHEMATICS 

of  making  him  manipulate  little  sticks  (very  simple  to 
procure)  as  much  as  possible,  and  reproduce  his  exercises 
on  a  slate  or  a  paper. 

We  are  now  going  to  enter  the  regions  of  higher  arith- 
metic. If  he  tends  to  be  puffed  up,  repress  this  display 
of  pride,  hinting  to  him,  first,  that  Algebra  is  one  of  the 
easiest  parts  of  Mathematics,  and  second,  that  he  does 
not  know  anything  and  is  not  learning  anything  now, 
except  games  which  will  be  useful  to  him  later,  when 
he  remembers  them. 


14.  We  begin  Algebra. 

Up  to  now  we  have  learned  to  add,  giving  the  sums,  and 
to  subtract,  giving  the  difference.  For  example,  the  sum 
of  8,  5,  and  14  is  27.  We  have  imagined  a  sign  or  symbol 
(  +  )  which  represents  addition,  and  which  expresses  phis, 
and  also  a  symbol  ( = )  which  expresses  equal  to.  So  that  in 
this  manner  the  result  which  we  have  just  recalled  might 
equally  be  written 

8  +  5  +  14  =  27 

and  would  read  8  plus  5  plus  14  equals  27. 

Similarly,  for  subtraction,  we  make  use  of  a  symbol  (— ) 
which  expresses  minus,  and  if  we  write  7  —  5  =  2,  that 
would  read,  7  minus  5  equals  2,  which  means  that  in 
subtracting  5  from  7,  we  obtain  2  as  the  difference. 

All  operations  of  this  nature  can  be  worked  by  means 
of  straws,  or  segments,  as  we  have  seen  before.  Thus, 
looking  at  Fig.  7  we  see  that  it  denotes 

5  +  3  +  4  =  12 

and  that  it  can  be  written  just  as  easily 
OA  +  AB  +  BC  =  OC 

Fig.  8  denote? 

11  —  4  =  7. 


WE   BEGIN  ALGEBRA  27 

We  can  amuse  ourselves  by  doing  this  in  whatever  manner 
we  like,  and  in  expressing  our  working  under  these 
different  forms. 

We  understand  that  in  the  place  of  8,  5,  14,  or  of  5, 
3,  4,  in  the  preceding  examples  we  could  put  any  other 
numbers  we  like ;  if  we  call  them  A,  B,  C,  writing  A  +  B 
+  C  =  S,  we  will  always  express  the  sum  of  three 
numbers ;  this  sum  would  be  27  in  the  first  example, 
12  in  the  second. 

In  the  same  manner  A  —  B  =  R  shows  that  the  differ- 
ence obtained  in  subtracting  B  from  A  is  equal  to  R. 
For  instance,  in  Fig.  8,  A  =  11,  B  =  4,  and  R  =  7. 

It  is  often  very  convenient  to  show  our  work  by  signs, 
and  to  replace  numbers  by  letters.  It  is  well  to  accustom 
ourselves  to  this  early,  because  it  will  be  most  useful  in 
the  future,  and  save  much  trouble.  We  must  also  know 
what  it  means  when  we  put  something  between  brackets 
thus 

(     )  +  (     )or(     )-(     ) 

This  simply  means  that  we  should  replace  each  bracketed 
term  by  the  result  which  it  gives.  For  instance, 

(A  -  B)  -  (C  -  D)  +  (E  -  F) 
if  A,  B,  C,  D,  E,  F, 

are  replaced  by          10,  2,  9,  6,  7,  5, 
would  express  (10  -  2)  —  (9  —  6)  +  (7  —  5), 
or  8—3  +  2,  that  is  to  say  7. 

All  these  ways  of  expression  are  sometimes  called 
algebraical.  But  the  words  themselves  are  of  little 
importance,  it  is  their  meaning  which  counts. 

What  follows  will  show  us  something  fresh.  When  we 
are  adding  numbers  we  can  go  on  indefinitely,  for  instance, 
with  several  heaps  of  beans  we  can  always  make  them 
into  a  single  heap.  In  other  words,  it  is  always  possible 
to  add,  and  we  can  express  the  addition  in  figures,  in 


28  MATHEMATICS 

counters,  in  matches,  in  little  sticks,  in  straws,  in  segments 
of  a  straight  line,  just  as  we  please. 

It  is  not  the  same  with  subtraction.  If  I  have  a  heap 
of  seven  counters,  for  instance,  from  which  I  wish  to 
take  away  10,  the  thing,  as  we  have  already  remarked, 
is  manifestly  impossible. 

However,  if  we  return  to  what  has  been  previously 
said,  as  shown  in  Fig.  8,  we  shall  be  obliged,  to  subtract 
by  means  of  straws,  or  of  seg- 
ments of  a  straight  line,  to  lay 
out  (Fig.  9)  on  a  straight  line 
a  segment  OB  equal  in  length 

to  7  matches,  then  from  B  lay  in  the  contrary  direc- 
tion, that  is  from  right  to  left,  a  segment  whose 
length  is  the  number  to  subtract ;  now  this  is  always 
possible ;  and  Fig.  9  demonstrates  it,  supposing,  as 
we  have  made  it,  that  the  number  to  subtract  is  10 ; 
we  obtain  thus,  the  length  BC  being  10,  a  point  C,  and  we 
have  for  remainder  the  segment  OC ;  only,  the  point  C 
is  no  longer  to  the  right  of  the  point  O  ;  it  is  to  the  left ; 
the  segment  OC  is  directed  from  right  to  left,  and  its 
length  is  equal  to  3. 

Such  a  number  is  said  to  be  negative ;  we  write  it 
. —  3,  we  call  it  "  minus  3  "  ;  and  it  would  be  correct  to  say 
7  -  10  =  -  8. 

This  creation  of  negative  numbers  makes  all  the  subtrac- 
tions possible  which  were  not  so  with  ordinary  numbers, 
which  we  call  by  contrast  positive  numbers. 

In  Fig.  10  all  the  part  to  the  right  of  point  O  represents 
the  domain  of  positive  numbers  (first  arrow) ;  all  the  part 
to  the  left  (second  arrow),  represents  the  domain  of  the 
negative  numbers ;  and  the  total  of  the  two  arrows, 
comprising  the  straight  line  in  its  entirety,  in  the  two 
directions,  represents  the  domain  of  Algebra. 

It  will  be  necessary  now,  when  we  wish  to  express 
numbers  by  straws,  or  segments,  to  pay  attention  to  the 
direction  of  these  segments,  or  to  the  sign  of  the  number  ; 


WE    BEGIN  ALGEBRA  29 

thus  (Fig.  9)  OB  will  be  a  positive  segment,  representing 
the  number  7  ;  OC  will  be  a  negative  segment  representing 
the  number  —  3,  itself  also  negative. 

This  compels  us  to  consider,  for  fear  of  making  an  error, 
which  of  the  two  ends  of  a  segment  we  will  call  the 
beginning,  and  which  the 

end  ;    and  the   direction  Negative  numbers^! — L»  Positive  numbers 
of  the   segment   will  be  0 

always  that  which  starts  Fl(J-  10- 

from  the  beginning  to  go  towards  the  end.  When 
we  write  the  segment  AB,  that  will  always  mean 
that  A  is  the  beginning  and  B  is  the  end.  We  shall 
be  obliged  to  change  slightly  the  appearance  of  our  little 
sticks.  It  will  be  quite  easy  to  blacken  them  slightly 
at  one  end  by  dipping  them  in  Indian  ink,  a  harmless 
dye ;  the  black  part  will  then  always  represent  the  end. 
So  that,  placing  three  matches  in  a  row,  the  black  end 
towards  the  right,  we  shall  express  the  number  +  3 ; 
placing  two  of  them  in  a  row,  the  black  end  to  the  left, 
the  number  —  2  is  shown,  and  so  on. 

It  will  always  be  correct  to  add  one  number  to  another 
by  laying  end  to  end,  in  the  proper  direction,  the  segments 
which  express  them.  For  instance,  to  add  11  and  —  4, 
we  will  take  a  segment  OB  of  the  length  of  11,  directed 
from  left  to  right,  and  afterwards  a  segment  BC  the 
length  of  4  directed  from  right  to  left.  Now  Fig.  8  is 
just  what  we  have  done  to  obtain  the  difference  11  —  4^ 
We  can,  therefore,  write  11  +  (  —  4)  =  11  —  4  =  7; 
and  subtractions  thus  take  us  back  to  additions. 

Exercises  on  the  negative  numbers  can  be  varied  as 
much  as  we  like,  and  will  be  quite  easy  to  do  with  sticks 
blackened  at  one  end.  We  can  also  make  straws  as  long 
as  several  sticks,  and  blacken  them  in  the  same  manner, 
to  show  which  is  the  end.  It  is  easy  to  get  accustomed  to 
this  simple  and  necessary  idea  of  the  sign  or  the  direction 
of  number. 

Besides,   if  the  negative  numbers  seem  puzzling  at 


30  MATHEMATICS 

first,  a  little  reflection  will  find  an  altogether  natural 
explanation.  At  the  first  blush  it  seems  as  if  there  could 
not  be  any  number  less  than  nothing,  that  is  zero. 
However,  in  ordinary  speech  we  say  every  day  that  the 
thermometer  registers  so  many  degrees  below  zero. 
When  we  wish  to  show  the  height  above  sea-level  of  any 
point,  we  understand  that  if  this  point  were  at  the  bottom 
of  the  sea,  it  would  be  below  zero. 

If  starting  from  home  I  want  to  calculate  the  distance 
that  I  shall  go  in  one  direction,  and  if  I  walk  in  exactly 
the  opposite  direction,  I  know  perfectly  well  that  I 
cannot  use  the  same  number  to  express  two  opposite 
things. 

A  man  without  any  fortune,  but  who  owes  nothing,  is 
not  rich ;  but,  if  he  has  no  fortune  and  has  debts,  we  can 
say  that  he  has  less  than  nothing  ;  his  fortune  is  negative. 

A  cork  has  a  certain  weight ;  if  we  throw  it  in  the  air, 
it  falls  ;  plunge  it  into  water,  and  let  it  go,  it  rises ;  its 
weight  has  become  negative,  in  appearance  at  least. 

Briefly,  negative  numbers,  far  from  being  mysterious 
in  their  character,  adapt  themselves,  in  the  most  natural 
fashion,  to  all  quantities,  and  there  are  some  which,  from 
their  nature,  can  be  measured  in  two  ways  opposed  to  each 
other,  such  as  hot  and  cold,  high  and  low,  credit  and  debit, 
future  and  past,  etc.  By  means  of  concrete  examples, 
we  can  make  these  simple  ideas  sink  into  the  mind  of 
very  young  children,  for  everything  we  have  said  is 
extremely  simple  and  easy  of  comprehension. 

The  pupils  will  be  interested  if  we  continually  accom- 
pany our  explanations  with  examples  carried  out  by  means 
of  sticks  and  straws,  and  that  will  be  more  profitable  for 
the  formation  of  their  minds  than  the  monotonous 
recitation  of  non-understandable  rules,  or  of  incompre- 
hensible definitions. 

They  have,  so  far,  only  practised,  by  means  of  games, 
the  two  first  rules  of  arithmetic ;  it  is  only  a  short  time 
since  they  were  learning  to  write  figures,  or  to  form  various 


CALCULATIONS;   MEASURES;    ETC.  31 

letters,  and  now  behold  them  plunged  headlong,  and  you 
with  them,  right  in  the  midst  of  Algebra.  If  you  mention 
this  formidable  word  before  them,  do  not  fail  to  tell  them 
that  this  science  which  is  so  useful  and  so  wonderful, 
is  comparatively  speaking  modern,  and  that  it  is  FranQois 
Viete  l  to  whom  the  honour  belongs  of  having  been  its 
inventor. 


15.  Calculations ;  Measures ;  Proportions. 

We  have  seen  from  the  beginning  that  what  we  are 
constantly  doing  is  to  calculate  and  to  measure.  If  we 
have  before  us  a  heap  of  grains  of  corn,  and  if  we  find 
upon  counting  them,  that  there  are  157,  this  number, 
as  we  have  previously  noticed,  would  be  useful  in 
representing  to  us  a  collection  of  counters,  of  matches,  of 
trees,  of  sheep,  or  in  short  anything.  If  to  determine 
length  we  have  put  sticks  that  are  all  alike  one  after 
another,  and  if  we  find  157  will  measure  this  length,  we 
say  that  it  is  the  same  length  as  157  sticks.  In  all  these 
various  cases  we  should  never  be  able  to  value  anything 
if  we  had  not  before  us  the  idea  of  a  grain  of  corn,  a 
counter,  a  tree,  a  sheep,  or  a  stick. 

Number  has  no  significance  except  by  the  comparison 
that  it  brings  about  with  the  single  object  (grain  of 
corn,  counter,  etc.),  without  which  it  would  be  impossible 
to  make  it,  and  this  single  object  is  called  "unity."  This 
comparison  is  what  we  term  a  proportion,  and  this  idea 
of  proportion  leads  us  on  to  say  that  a  number  is  simply  a 
proportion  of  the  number  to  unity. 

It  is  all  the  more  necessary  to  fix  this  firmly  in  the  mind 
of  the  pupil,  because  unity  is  not  always  the  same. 
For  instance,  having  formed  bundles  of  sticks,  let  us  take 
a  heap  and  count  them ;  we  find  there  are  7 ;  seven  is 

1  Victe,  French  mathematician,  born  at  Fontenay-le-Compte  (1540 — 
1603). 


32  MATHEMATICS 

the  proportion  of  our  collection  of  sticks  to  one  bundle, 
which  is  unity.  Now,  let  us  scatter  our  sticks  by 
unfastening  the  bundles,  and  let  us  count ;  it  is  the  stick 
which  now  becomes  unity,  and  we  can  count  seventy  of 
them  ;  this  number  will  be  the  proportion  of  the  collection 
tc  one  stick. 

Similarly,  let  us  take  three  faggots  of  sticks  ;  if  we  count 
by  bundles,  we  shall  find  thirty  bundles  ;  but  if  by  sticks, 
three  hundred. 

Three  will  be  the  proportion  of  the  whole  heap  of  sticks 
to  a  faggot ;  thirty  the  proportion  of  the  same  heap  to 
a  bundle ;  three  hundred  the  proportion  to  one  stick. 

We  might  give  innumerable  instances  of  such  examples, 
varying  them  indefinitely,  in  such  a  manner  as  to  make 
this  idea  of  proportion  perfectly  familiar  to  the  pupil. 
This  is  the  root  of  all  calculation  and  all  measurement, 
but,  by  some  strange  hallucination,  in  academical  teach- 
ing it  is  put  at  the  end  of  arithmetic.  It  is  not 
possible  to  count  two  beans  without  having  this  idea 
of  the  proportion  of  two  to  one  ;  nor  to  measure  a  length 
of  three  yards  without  comparing  the  length  with  that 
of  one  yard  (proportion  of  three  to  one),  and  so  on. 

At  this  stage  it  will  be  desirable  to  show  the  pupil, 
without  any  theoretical  explanation,  without  any  defini- 
tion, without  any  appeal  to  his  memory,  the  commonest 
measures,  weights  and  coins  which  we  find  ready  to  our 
hand,  yards,  quarts,  ounces,  cents,  etc. 

We  will  give  him  exercises  in  making  use  of  them, 
accustoming  himself  to  them  to  measure  and  to  count, 
and  the  idea  of  proportion  will  insensibly  grow  in  his 
mind,  will  associate  itself  indissolubly  with  that  of  number, 
which  is  essential  for  the  day  in  the  future  when  he  will 
pass  from  play  to  work.  And  this  work  can  become  not 
only  interesting  but  amusing,  instead  of  being  a  wearisome 
task,  if  not  a  torture. 


THE  MULTIPLICATION  TABLE 


33 


16.  The  Multiplication  Table. 

We  are  now  going  to  learn  how  to  make  a  little  table 
which  will  be  most  useful  to  us  because  of  what  follows, 
and  will  be  a  very  good  exercise,  even  for  its  own  sake. 
Under  the  form  as  represented  below,  this  table  is  generally 
called  the  table  of  Pythagoras,1  which  may  have  been 
invented  by  this  wonderful  man,  although  it  is  not  by  any 
means  certain  ;  however,  even  its  reputed  origin  proves 
to  us  that  it  is  not  anything  new. 

To  form  the  multiplication  table  we  begin  by  writing 
on  a  sheet  of  squared  paper  the  first  nine  numbers  in  the 
nine  squares  which  follow  each  other : — 

123456789. 

Then,  taking  the  first  figure,  1,  we  add  it  to  itself,  which 
makes  2,  which  we  write  below;  then  1  to  2,  which  makes  3; 
and  so  on,  which  gives  the  first  column  of  Fig.  11. 

We  will  do  the  same  to  fill 
the  other  columns,  but  the 
important  thing  is  to  write  the 
results  only  and  nothing  else. 
For  example,  for  the  column 
which  begins  with  7,  we  would 
say,  "7  and  7,  14  ;  and  7,  21 ; 
and  7,  28  ;  and  7,  35  ;  and  7, 
42 ;  and  7,  49 ;  and  7,  56 ; 
and  7,  63."  And  we  write  in 
succession  14,  21,  28,  ...  63 
in  the  column  beginning  with  7.  FIG.  11. 

It  will  be  sufficient,  we  can  see,  for  the  pupil  who  knows 
his  addition  table,  to  be  able  to  form  this  table  very 
quickly.  When  he  has  completed  it,  we  see  that  the  rows 
and  the  columns  are  all  alike.  Thus  the  row  which  begins 
with  3  contains,  like  the  column  beginning  with  3,  the 
numbers  3,  6,  9,  ...  27. 

1  Pythagoras,  Greek  philosopher,  born  at  Samoa,  6th  century  B.C. 
M.  D 


1 

2 

3 

4 

S 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

8 

12 

16 

20 

24 

28 

32 

36 

5 

10 

15 

20 

25 

30 

35 

40 

45 

6 

12 

18 

24 

30 

36 

42 

48 

54 

7 

14 

21 

28 

35 

42 

49 

56 

63 

8 

16 

24 

32 

40 

48 

56 

64 

72 

9 

18 

27 

36 

45 

54 

63 

72 

81 

34  MATHEMATICS 

It  is  absolutely  necessary  to  plant  this  table  firmly  in 
our  minds.  But  the  proper  way  to  arrive  at  this  is  not 
to  attempt  to  learn  it.  This  is  accomplished  by  making 
it,  verifying  it,  by  carefully  examining  it,  making  use  of 
it  as  we  will  show  later  on.  If  it  does  not  come  readily 
to  the  mind,  the  child  must  re-construct  it — not  a  very 


FIG.  12.     . 

long  business ;  and  then  he  will  finish  by  seeing  it  with 
his  eyes  shut. 

We  could  make  out  the  table  beyond  9,  but  this  is  not 
advisable  ;  because  if  we  go  on,  for  instance,  up  to  20  or 
25,  it  will  take  much  longer  to  do,  and  it  is  not  necessary 
for  the  child  to  remember  the  table  written  out  so  far, 
although,  of  course,  it  would  be  useful. 

He  will  notice  certain  peculiarities  about  this  table. 
Thus  in  the  column  (or  the  row)  beginning  with  5,  the 


PRODUCTS  35 

figures  of  the  units  are  alternately  5  and  0 ;  in  the  column 
(or  the  row)  beginning  with  9,  the  figures  of  the  units 
8,  7,  6,  ...  always  diminish  by  1,  and  those  of  the  tens 
1,  2,  3,  .  .  .  increase  by  1.  The  explanation  of  this  will 
not  be  difficult  to  find. 

It  is  worthy  of  notice  that  we  can  make  a  multiplication 
table  without  using  a  single  figure  ;  to  do  this,  it  is  only 
necessary  to  have  a  squared  paper  in  sufficiently  small 
divisions  (Fig.  12).  The  table  shown  in  this  figure  is 
made  out  as  far  as  10.  We  proceed  as  follows  :  mark 
divisions  out  successively,  on  a  horizontal  line,  1,  2,  3, 
...  10,  and  mark  the  points  of  division.  Then,  on  a 
vertical  line,  taking  the  same  starting  point,  do  the 
same  thing ;  mark  off  the  points  of  division  by  means 
of  strong  lines,  and  we  obtain  large  divisions ;  and  each 
of  these  divisions  contains  a  number  of  little  ones  which 
will  be  precisely  the  same  as  is  contained  in  our  table  of 
figures.  The  explanation  of  this  is  quite  simple ;  for 
our  table  in  Fig.  12  only  represents  by  means  of  a  series 
of  lines  what  in  Fig.  11  is  done  by  calculation. 

17.  Products. 

If  I  take  a  heap  of  7  sticks,  and  I  form  three  such  heaps, 
I  can  ask  myself,  "  How  many  sticks  will  there  be  in  ah1?  " 
That  is  called  multiplying  7  by  3.  The  result  obtained 
by  such  multiplication  is  the  product  of  the  two  numbers 
7  and  3;  7  is  the  multiplicand,  and  3  the  multiplicator. 
If,  instead  of  heaping  up  the  sticks,  we  leave  the  three 
heaps  separate,  we  see  that  (taking  one  heap  as  repre- 
senting a  unit)  the  number  showing  the  product  will  be 
3  ;  or  that  the  proportion  of  the  product  will  be  3  ;  which 
number  also  gives  us  the  proportion  of  3  to  1. 

So  we  can  say  equally  well  that  to  multiply  7  by  3  is 
to  repeat  7,  3  times ;  or  to  find  a  number  of  which  the 
proportion  to  7  will  be  the  same  as  that  of  3  to  1. 

To  multiply  a  number  (the  multiplicand)  by  another 

D2 


36  MATHEMATICS 

(the  multiplicator)  is  to  find  another  number  (the  product), 
which  may  be  formed  by  repeating  the  multiplicand  as 
many  times  as  there  are  units  in  the  multiplicator ;  this 
product,  in  short,  bears  the  same  proportion  to  the 
multiplicand  which  the  multiplicator  does  to  unity. 

These  are  not  formulae  which  the  child  must  learn ; 
they  are  ideas  which  he  must  insensibly  assimilate  by 
our  aid  ;  for  the  former  present  an  appearance  of  difficulty, 
while  the  latter  can  be  easily  grasped  by  the  pupil, 
especially  if  we  take  the  trouble  of  expressing  them  by 
means  of  grains  of  corn,  of  sticks,  or  of  divisions  on 
squared  paper. 

It  is  quite  certain  that  the  child  will  jump  immediately 
to  the  idea  that,  to  find  a  product,  he  has  nothing  to  do 
but  make  an  addition,  that  the  product  of  7  by  3  is 
7  +  7  +  7  in  the  same  way  that  3  is  1  +  1  +  1.  And 
as  the  table  (Fig.  11)  has  been  made  in  this  way,  it  gives 
us  the  desired  product  by  taking  the  column  that  begins 
with  7,  the  line  which  begins  with  3  and  looking  for  the 
meeting  point,  where  we  read  21. 

Explain  at  this  point  that  the  multiplication  sign  is  X , 
and  that  therefore  the  phrase  "  the  product  of  7  multiplied 
by  3  is  21  "  is  expressed  thus  :  7  x  3  =  21. 

7  X  3  is  often  written  7.3 ;  instead  of  these  two 
we  can  have  any  two  numbers  whatever  represented 
by  the  letters  a,  b.  Their  product  can  be  expressed  by 
a  x  b,  or  a.b,  or  simply  db  ;  when  we  write  ab  =  p,  we 
mean  that  a  multiplied  by  b  is  p. 

It  is  well  to  know  also  that  we  can  consider  products 
such  as  a  x  b  x  c  x  d,  or  simply  abed,  for  instance ;  this 
means  that  we  multiply  a  by  b,  then  the  resulting  product 
by  c,  then  the  new  product  by  d ;  a,  b,  c,  d,  are  called  the 
factors  of  the  product  abed  ;  we  can  thus  have  the  product 
of  any  number  of  factors. 

As  regards  the  practice  of  multiplication,  we  must 
first  notice  that  the  table  gives  us  the  answers  when  the 
multiplicand  and  the  multiplicator  are  each  less  than  ten. 


PRODUCTS 


37 


It  will  be  easy  afterwards  to  show  how  we  can  multiply 
a  number  by  10,  100,  1,000. 

The  employment  of  numerous  examples  conforming  to 
all  the  rules  given  in  all  the  arithmetic  books  might  be 
useful,  provided  that  it  is  not  accompanied  by  any  theory. 
I  cannot  sufficiently  impress  upon  you  the  importance 
of  giving  preference  to  the  Mohammedan  method,  which 
is  almost  as  quick,  much  easier  to  understand  and  carry 
out,  and  not  sufficiently  known  in  teaching  although 
approved  by  several  writers. 

We  are  going  to  take  the  very  simple  instance  of 
9,347  x  258.  The  multiplicand  has  4  figures,  and  the 
multiplicator  has  3 ;  let  us  mark 
out  on  squared  paper  3  rows 
of  4  divisions  each  ;  on  top  of 
this  figure  let  us  write  the 
figures  of  the  multiplicand  9, 
3,  4,  7,  from  left  to  right ;  at  the 
left  and  working  from  the  bottom 
to  the  top,  those  of  the  multi- 
plicator 2,  5,  8  ;  having  traced 
the  dotted  lines  of  the  figure, 
let  us  now  put  in  each  division  the  product  of  the  two 
corresponding  numbers,  as  though  we  were  making  a 
multiplication  table,  but  always  placing  the  figure  of  the 
tens  of  the  product  below,  and  that  of  the  units  above 
the  dotted  line ;  finally,  we  add  up,  taking  for  the 
direction  of  the  columns  the  dotted  lines ;  thus  we  find 
the  product  2,411,526.  The  great  advantage  of  this 
method  is  that  it  does  not  necessitate  any  partial 
multiplication  nor  the  observance  of  any  special  order. 
As  all  the  divisions  are  filled  we  are  certain  to  forget 
nothing. 

With  the  same  example,  and  the  same  method,  we 
indicate  a  slightly  different  arrangement  which  does  not 
compel  us  to  add  up  obliquety,  and  which  in  this  sense 
is  perhaps  most  convenient  (Fig.  14). 


K 

H 

H 

H 

V5 

>-5 

K 

K 

s 

\6 

-\8 

M 

2411 

FIG.  13. 

38  MATHEMATICS 

As  regards  the  justification  of  the  Mohammedan  method, 
it  is  sufficiently  evident  to  everyone  acquainted  with  the 
theory  of  multiplication,  although  at  present  unnecessary 
for   the    child.     If  he    is    of    an 
enquiring    disposition     he    will 
probably  discover  it  for  himself. 
What  is  of   real   importance    is 
that  he  shall  be  able  to  calculate 
correctly,    and    that    he    will   be 
interested  in  doing  so.     From  the 
moment  when    fatigue    or   bore- 
2   4-"  1    1    526        dom  overtakes  him  it  is  absolutely 
necessary  to  go  on  to  something 

A*  I0r.      14.  1 

else. 

However,  we  will  not  abandon  what  relates  to  multipli- 
cation without  recalling  that  a  product 

a  x  a  x  a  x   ...    x  a, 

of  which  all  the  factors  are  equal,  is  called  a  power  of  a ; 
that  such  a  product  is  written  a",  n  being  the  number  of 
the  factors,  and  that  we  call  it  the  nth  power  of  a  ;  that 
the  second  power  is  called  square,  and  the  3rd  cube 
(we  shall  soon  see  the  reason  for  this).  The  number  n 
is  called  the  index. 

2  x  2  x  2  x  2  =  24  =  16 ;  the  index  is  4. 

The  cube  of  5  is 

5x5x5  =  53  =  125  ;   the  index  is  3. 
The  square  of  7  is 

7  x  7  =  72  =  49  ;  the  index  is  2. 


18.  Curious  Operations. 

There  are  a  number  of  results  from  operations  which 
strike  us  because  of  their  peculiarities.  Their  great  merit 
consists  in  arousing  the  child's  curiosity,  and  thus  giving 
him  a  taste  for  calculation. 


CURIOUS  OPERATIONS  39 

Subjoined  are  various  examples  which  will  be  sufficient 
for  our  purpose. 

I.  Ask  the  child,  after  giving  him  a  sealed  envelope, 
to  write  a  number  of  3  figures ;   let  him  choose  one  to 
suit  his  own  fancy.     Suppose  he  writes  713  ;    then  write 
it  the  reverse  way,  which  makes  it  317 ;    next,  subtract 
one  from  the  other,  which  gives  the  result  396  l ;  reverse 
this  number,  which  then  reads  693 ;  finally  add  these  last 
two,  which  will  be  1089.     At  this  point  ask  him  to  open 
the  envelope  ;  he  will  find  in  it  a  paper  on  which  you  have 
written  beforehand  1089.     It  is  remarkable  that  any  other 
number  of  3  figures  would  have  led  to  the  same  result, 
provided  that  the  first  and  third  figures  are  different.-' 

II.  If  the  child  works  out  12  x  9  +  3,  123  x  9  +  4, 
and  so  on,  as  far  as  123456789  x  9  +  10,  each  result  will 
contain  no  other  figure  than  1  written  a  varying  number 
of  times,  12  x  9  +  3  giving  us  as  result  111,  and  123  x  9 
+  4  equalling  1111,  and  so  on. 

On  the  contrary,  working  out  9  x  9  +  7,  98  X  9  +  6, 
and  so  on  till  we  reach  9876543  x  9  +  1,  we  shall  have 
results  which  will  contain  no  figure  but  8. 

III.  The  product  of  123456789   X  9  will  be  a  number 
made   up   of   1's.     Taking  the   same   multiplicand   and 
multiplying  it  by 

18  (  =  9  X  2),  27  (  =  9  x  3)  ...  to  81  (=  9  X  9), 

we  shall  find  some  curious  products,  all  made  up  of  the 
same  figure  repeated. 

IV.  Take    the    number    142857 ;     if    we    multiply    it 
successively  by  2,  3,  4,  5,  6  our  products  will  be 

285714,     428571,     571428,     714285,     857142. 

It  will  be  seen  that  each  product  contains  the  same  figures 
as  the  multiplicand. 

1  This  difference  ought  always  to  have  three  figures.  If  there  are 

only  two,  we  must  put  a  nought  in  the  hundreds  place.  For  example, 

716  and  617  will  give  us  for  difference  099;  according  to  this  rule, 
adding  099  and  990  we  make  1089. 


40  MATHEMATICS 

Multiplying  it  by  7,  we  have  999999,  and  if  we  cut  it  in 
two,  making  142  and  857,  the  sum  of  these  numbers  will 
be  999.  The  same  result  is  obtained  by  cutting  in  two 
any  of  the  five  products  written  above. 

V.  Complete  these  examples  by  showing  the  easy 
method  of  multiplying  by  9,  by  99,  999.  This  should  always 
be  done  by  multiplying  by  10,  100,  1000,  etc.,  and  then 
subtracting  the  multiplicand.  If  the  numbers  are  not 
too  big  this  can  quite  soon  be  done  mentally. 

19.  Prime  Numbers. 

Running  our  eyes  over  a  multiplication  table,  we  notice 
that  it  includes  certain  numbers  up  to  the  limit  to  which 
it  extends,  but  not  all  these  numbers.  In  other  words, 
there  are  numbers  which  are  products,  and  others  which 
are  not ;  the  latter  are  called  the  prime  numbers,  the 
others  are  called  composite  numbers. 

For  instance,  2,  3,  5,  7,  29,  71  are  prime  numbers ; 
4,  6,  9,  87,  91,  are  composite  numbers,  because  4  =  2x2, 
6  =  2  X  3,  9  =  3  x  3,  87  =  3  x  29,  91  =  7  X  13.  How- 
ever far  we  advance  in  the  series  of  numbers  we  always 
meet  these  prime  and  composite  numbers.  This  distinc- 
tion is  fundamental ;  and  yet,  despite  the  labours  of  our 
most  learned  men,  we  know  very  little  about  prime 
numbers.  We  are  incapable,  if  the  number  under  con- 
sideration is  rather  large,  of  saying  whether  it  is  prime 
or  not,  unless  we  give  ourselves  up  to  a  groping  or  tentative 
method  which  requires  very  long  and  laborious  calcula- 
tions. This  shows  us  how  very  little  progress  science 
has  made  on  these  questions  which  appear  quite  simple, 
and  how  modest  we  ought  to  be  when  we  compare  the 
slight  extent  of  our  knowledge  with  the  immensity  of 
the  things  about  which  we  are  ignorant. 

Still,  from  antiquity  we  have  possessed  a  method  of 
finding  the  prime  numbers  to  a  limit  as  distant  as  we 
wish.  This  method  consists  in  writing  out  the  complete 


PRIME  NUMBERS  41 

list  of  these  numbers,  then  cancelling  those  which  are 
products  of  2,  of  3,  of  5,  etc.  We  intend  applying  it  here 
to  the  first  150  numbers.  Write  them,  suppressing  1, 
which  is  useless. 


3     -V     5  Jf  7  -8"  •#"  -tff    II    -tt     13 

-W    !7  -W  i9  -W  -frf  -W   23   -«r  - 

-38-    29  -dO-  31  -32-  -*3-  -34*  -#T  .30-   37 

41  -W  43  -4#--4S~>W    47 

2-  53  -£*•  -^r  -^e^^r  -s*r  59 

67  -&tf-.-#r-Kr  7i  j?i-  73 
79  -mr-sr-sr  83  ^r  -et 
89  -^r  ^r  -^r  -^3-  -wr  -»tf  -flfr-  97 
.m-  103  -f^r-twr^^Hr  107  4wr  109 

r  113  44^ 

127  .*•*«•  4-*ir-4^(T  131 
»«-  1:17 
149 


Starting  from  2,  and  going  on  by  2's,  according  to  t  he- 
list,  we  find  the  products  of  2,  which  are  4,  6,  .  .  .,  that  is 
to  say,  even  numbers  ;  they  are  not,  therefore,  prime 
numbers,  and  we  cancel  them  at  one  blow  as  far  as  150. 

Let  us  begin  this  time  with  3  and  continue  by  3's, 
cancelling  the  numbers  which  we  find  to  be  products  of 
multiplication  by  3,  unless  they  have  been  cancelled 
already. 

The  first  number  uncancelled  after  3  is  5  ;  starting 
therefore  from  5,  and  moving  on  by  5's,  we  proceed  in 
the  same  way.  Doing  the  same  with  7  and  11,  we  find 
that  only  the  following  numbers  remain  uncancelled  :  — 

2  3  5  7  11  13  17  19  23  29  31  37 
41  43  47  53  59  61  67  71  73  79  83  89 
97  101  103  107  109  113  127  131  137  139  149 

which  gives  us  the  list  of  prime  numbers  up  to  150. 


42  MATHEMATICS 

This  ingenious  process  is  known  as  the  sieve  of 
Eratosthenes1  from  the  name  of  the  inventor. 

20.  Quotients. 

Some  small  heaps  of  counters,  each  containing  7,  have 
been  formed  into  one  large  one,  which  contains  56.  We 
want  to  find  out  how  many  little  heaps  we  have  used  to 
make  the  large  one. 

What  we  are  now  going  to  do  to  discover  this  is  called 
a  division.  We  can  also  say  that,  knowing  the  product  of 
two  factors,  56,  and  one  of  the  factors,  7,  we  wish  to  find 
the  other. 

The  product  given,  i.e.,  56,  is  called  the  dividend ;  the 
factor  given,  i.e.,  7,  is  called  the  divisor,  and  the  result 
we  are  seeking  is  the  quotient. 

We  really  could  find  the  quotient  by  subtraction,  that 
is  to  say,  by  taking  the  divisor  from  the  dividend,  then 
the  divisor  from  the  remainder  obtained,  and  so  on, 
until  there  is  nothing  left,  being  careful  to  count  how  many 
subtractions  we  have  made.  Thus,  successively  taking 
away  7  from  56,  we  have  the  numbers  49,  42,  35,  28,  21, 
14,  7,  and  we  see  that  we  have  needed  to  subtract  8  times 
to  exhaust  our  dividend  56.  The  quotient  we  want  is 
thus  8 ;  naturally,  our  knowledge  of  the  multiplication 
table  would  show  us  this  immediately. 

The  method  of  successive  subtractions  would  be 
impracticable  with  rather  large  numbers,  because  of  its 
length.  The  classical  rule  that  is  adopted  in  doing 
division  is  nothing  else  than  a  means  of  counting  very 
much  faster  subtractions  done  all  at  once. 

To  accustom  children  to  be  able  to  divide  with  ease,  we 
must  begin  by  always  making  them  form  into  a  little 
table  the  products  of  the  divisor  by  2,  3,  ...  9.  This 
does  away  with  the  hesitation  that  so  hampers  the 
beginner  in  all  his  work. 

1  Eratosthenes,  Alexandrian  scholar,  born  at  Cyrene  (276 — 193  B.C.). 


QUOTIENTS  43 

We  will  show  him  how  this  is  done  by  the  example 
given  below  of  the  division  of  643734  by  273.  The 
products  of  273  are 

1  X  273  =    273  6  X  273  =  1638 

2  =546  7  =  1911 

3  =819  8  =  2184 

4  =  1092  9  =  2457 

5  =  1365 

Let  us  work  out  the  problem  in  the  ordinary  way: — 
643734  (273 
546          2358 


977 
819 

1583 
1365 

2184 
2184 

0 

In  the  dividend  there  are  643  thousand  ;  as  our 
little  table  shows  us,  643  contains  the  divisor  (273) 
twice;  therefore,  taking  away  546  thousand  from  the 
dividend,  we  make  2  thousand  subtractions  at  once. 
We  have  left  97734,  which  contains  977  hundreds  ;  977 
contains  the  divisor  3  times,  so  taking  away  819  hundreds 
we  have  made  300  subtractions  at  once.  Now  our 
remainder  is  15834,  containing  1583  tens  ;  1583  contains 
the  divisor  5  times,  and  taking  away  1365  tens,  we  make 
this  time  50  subtractions.  Finally,  there  remains  2184, 
which  contains  the  divisor  exactly  8  times  ;  therefore, 
by  subtracting  8  times  more  we  shall  have  taken 
everything  from  the  dividend,  and  the  quotient  will  be 
2358,  the  total  number  of  subtractions. 
The  child  should  form  the  habit  of  dividing  in  this  way 


44  MATHEMATICS 

without  it  being  necessary  to  explain  in  great  detail  what 
we  have  just  expressed  at  length. 

The  above  division  could  be  done  because  the  divisor 
and  dividend  had  been  picked,  but  if  we  take  two  numbers 
at  random,  it  is  hardly  likely  such  division  will  be 
possible.  We  will  not  be  able  to  take  away  the  divisor 
a  certain  number  of  times,  so  that  nothing  is  left. 
However,  if  we  work  on  precisely  the  same  lines,  taking 
the  divisor  from  the  dividend  as  many  times  as  we  can, 
we  shall  come  at  last  to  a  point  when  the  dividend  will 
be  smaller  than  the  divisor.  This  number  is  termed 
the  remainder  of  the  division. 

As  a  very  simple  example,  let  us  suppose  that  we  are 
going  to  divide  220  by  12.  We  shall  discover  this  to  be 
impossible,  for  when  12  "has  been  taken  18  times  from 
220,  4  will  remain  ;  it  follows  that  220  —  4,  or  216,  would 
be  divisible  by  12,  and  the  quotient  would  be  18.  These 
impossible  divisions,  when  we  get  a  remainder,  can  easily 
be  turned  into  divisions  which  are  possible,  by  simply 
replacing  the  dividend  by  tliis  dividend  with  the  remainder 
taken  from  it. 

Do  not,  however,  lay  much  stress  on  this  operation  of 
division,  except  from  the  point  of  view  of  the  practice  of 
calculation.  Theories  are  interesting,  but  will  be  more 
useful  at  a  later  period.  They  are  hardly  suitable  for 
the  introductory  period. 

It  is  useful  to  know  that  division  is  shown  by  a  sign 

56 
-r-  or  — -1      Thus  56  4-  7  or  y  expresses   the  quotient 

56 

of  56  by  7.     We  can  write  56  -r-  7  =  —  =  8.     In  general, 

r-  =  q  means  that  the  quotient  of  the  division  of  a  by 
b  is  the  number  q. 

1  As  the  minus  sign  and  the  division  sign  (2nd  method)  are  similar, 
it  \vould  be  well  to  point  out  that  the  minus  sign  is  placed  between 
two  numbers,  one  following  the  other,  whereas  the  division  sign  is 
placed  between  two  numbers,  one  placed  above  the  other.  [TB.] 


THE  DIVIDED  CAKE;  FRACTIONS          45 


21.  The  Divided  Cake;  Fractions. 

Suppose  five  people  wish  to  share  a  round  cake  equally. 
It  will  have  to  be  cut  into  five  perfectly  equal  pieces 
(Fig.  15)  by  incisions  starting  from  the  middle,  and  one 
of  these  pieces,  as  AOB,  will  be  the  share  of  each  one  of 
the  five.  This  part  is  called  a  fifth  of  the  cake,  and 

AOB  will  be  represented  by  —  of  a  cake. 

0 

Of  the  five  people,  two  may  be  absent,  but  their  shares 
are  to  be  reserved  for  them,  so  the  parts  AOB,  BOC, 
both  exactly  alike,  are  put  on  one  side 
for  them.  These  two  pieces  taken 
together  are  called  two-fifths  of  a  cake, 
and  they  are  shown  by  the  figures 

2  2 

•=.    All  these  numbers,  such  as  ^,  are 

5  5 

called  fractions;  2  and  5  are  the  two 
terms ;  2,  which  is  above  the  line,  is  the 
numerator,  indicating  the  number  of 
pieces;  5,  which  is  below,  is  the  denominator,  showing 

into  how  many  equal  pieces  we  have  divided  the  cake. 
5 

If  we  had  taken  -  of  the  cake,  we  can  see  that  would 
0 

have  been  the  whole  cake  ;  and  if  the  cake  had  been 
divided  into  any  number  whatever  of  equal  parts,  by 
taking  afterwards  the  same  number  of  parts  the  cake 

would  be  again  entire,  so  that  the  fraction  —  of  which 

the  numerator  and  denominator  are  alike  is  always  equal 
to  1. 

But  suppose  10  people,  and  not  five,  wanted  to  divide 
the  cake ;  then  we  should  have  had  to  divide  the  cake 
into  10  equal  parts,  and  the  tenths  could  be  obtained 
by  taking  fifths,  such  as  AOB,  and  cutting  each  of  them 

2        1 

into  two  equal  parts.    Then  TQ  =  5  J  and  it  is  no  harder 


46  MATHEMATICS 

to  see  that  two  fractions  are  equal  if  we  can  pass  from  one 
to  another  by  multiplying  the  two  terms  by  the  same 
number.  This  fundamental  principle  of  the  whole 
theory  of  fractions  is  proved  in  this  manner  by  a  sort  of 
intuitive  evidence,  by  means  of  concrete  objects,  and  we 
need  ask  for  no  demonstration. 

Suppose  now,  there  are  17  cakes,  all  alike,  and  that  5 
people  want  to  divide  them  equally.  There  are  two  ways 
of  doing  this.  We  can  divide  each  cake  into  fifths,  and 

17 

each  person  can  take  a  fifth  of  each  cake,  in  ah1   -z-,  or, 

secondly,  we  can  give  each  one  an  equal  number  of  cakes, 
which  of  course,  for  five  people  will  be  3,  as  5  is  contained 
in  17  three  times.  Then  we  shah1  be  left  with  2  cakes 
to  divide.  Dividing  them  into  fifths,  each  of  the  five 

2  17 

persons  will  take  -= ;  by  this  means  we  see  that  -jr  =  3 

2 

+  5' 

It  is  easy  by  this  means  to  initiate  the  child  into  all 
the  ordinary  fractional  calculations,  on  which  it  seems 
unnecessary  to  dilate  longer.  But  we  can  only  succeed 
by  always  using  concrete  objects,  such  as  cakes,  apples, 
oranges,  divided  lengths,  etc.  Then  he  will  seize  the  idea 
wonderfully  well,  that  these  new  arithmetical  expressions 
arc  numbers,  and  that  they  express  proportions. 

We  ought  at  this  stage  to  make  it  clear  that  these 
numbers  can  only  be  used  in  relation  to  those  things 
which,  by  their  very  nature,  are  divisible,  such  as  those 
we  have  indicated  above  ;  if,  for  example,  a  question 
involving  the  consideration  of  a  certain  number  of  persons 
was  placed  before  us,  the  application  of  fractional  numbers 
would  be  absurd,  and,  as  the  result  would  show,  impossible. 
It  is  to  be  regretted  that  this  is  not  more  frequently 
mentioned  to  the  pupil. 

In  other  words,  calculations  adapt  themselves  to 
suitable  things,  which  are  very  numerous,  but  not 


THE  DIVIDED   CAKE;   FRACTIONS  47 

universal.  And  this  further  maxim  must  always  be  borne 
in  mind  by  the  child,  that  in  arithmetic  there  is  a  necessity 
for  the  exercise  of  his  own  reflection  and  common  sense, 
without  which  a  mere  dexterity  in  computation  will  be 
useless. 

For  instance,  here  is  a  problem,  particularly  mentioned 
by  Edward  Lucas,1  which  will  serve  as  a  useful  exercise 
on  this  maxim.  A  tailor  has  a  piece  of  stuff  16  metres 
in  length,  from  which  he  cuts  off  2  metres  each  day ; 
in  how  many  days  will  he  have  cut  up  the  whole  piece  ? 
A  want  of  thought,  joined  to  the  habit  of  mechanical 
calculation,  leads  to  the  answer  8,  instead  of  the  number 
7,  which  an  exercise  of  common  sense  would  indicate. 

Questions  involving  fractions  should  be  varied,  not 
too  complicated,  and  borrowed  from  effective  concrete 
subjects.  It  would  be  wise  to  complete  them  by  drawing 
attention  to  decimal  fractions,  to  the  manner  in  which 
they  may  be  written,  and  to  the  methods  of  calculation 
connected  with  them. 

Many  good  treatises  on  arithmetic  will  furnish  the 
necessary  instances  in  this  respect.  I  content  myself 
with  insisting  on  the  usefulness,  above  all,  of  measures 
of  length,  and  instances  taken  from  the  counting  of 
money. 

Finally,  we  must  notice  that  if  the  sign  of  division  and 
notation  of  fractions  are  alike,  it  is  not  by  chance,  and 

15 

does  not  give  rise  to  confusion ;  -~  ,  for  instance,  expresses 

not  only  the  quotient  of  the  division  of  15  by  3  but  the 

15 

fraction  -^.    This  can  be  seen  with  concrete  objects,  and 

we  only  need  mention  it. 

The  properties  and  calculation  of  fractions  can  also  be 

1  Edward  Lucas,  French  mathematician,  born  at  Amiens  (1842 — 1891). 
He  was,  perhaps,  the  man  who,  in  his  day,  understood  better  than  any 
other  the  science  of  numbers.  His  merit  has  been  singularly  unrecog- 
nised, and  this  contributed  to  his  premature  death. 


48 


MATHEMATICS 


showr  *n  a  very  simple  fashion,  by  making  use  of  squared 
paper. 

Judgment  can  be  passed  upon  it  by  the  remarks  which 
follow.  They  only  presented  themselves  to  my  mind 
after  the  publi cation  of  the  second  edition.  I  will  endeav- 
our to  explain  them  now,  as  briefly  as  is  consistent  with 
a  sufficient  clearness  of  expression.  I  addv-  s  myself 
to  teachers,  and,  provided  they  have  understood  what 
I  have  said,  they  will  have  no  trouble  in  using  the  means 
proposed,  under  whatever  form  seems  good  to  them — 
that  is,  of  course,  if  they  agree  with  the  principle  of  my 
point  of  view. 

Suppose,  then,  we  represent  a  concrete  unit  of  any  kind 


FIG.  16. 


FIG.  17. 


(provided  that,  by  its  nature,  it  is  divisible)  by  means 
of  a  rectangle  (Figs.  16,  17). 

If  we  divide  the  length  of  this  rectangle  into  5  equal 
parts,  we  can  cut  it  into  5  portions,  into  5  vertical  bands 
all  alike.  Each  of  them  will  be  a  fifth  of  the  unit  (Fig.  16). 

If  we  divide  the  height  of  the  ectangle  into  5  equal 
parts,  we  can  also  cut  it  into  5  1  jrizontal  bands,  each 
being  alike,  and  each  of  them,  also,  a  fifth  of  thi  unit 
(Fig.  17). 

Again,  divide  the  length  (Fig.  18)  into  3  equal  parts, 
and  the  height  into  4  such  parts,  the  unit  can  be  cut  by 
lines  passing  through  the  points  of  division  into  12,  or 
3x4  little  rectangles,  all  alike,  and  each  being  a  twelfth 
of  the  unit. 

Taking  any  number  whatever  of  these  bands  or  rect- 


TIIE  DIVIDED   CAKE;  FRACTIONS 


49 


angles,  we  have  what  is  called  a  fraction.  If  the  number 
of  bands  or  little  rectangles  is  less  than  those  which  make 
up  the  unit  we  have  a  fraction  properly  so  called,  or  a 
proper  fraction.  If  greater,  we  have  a  fractional  expres- 
sion or  an  improper  fraction.  Therefore  a  proper  fraction 
is  less  than  1,  and  an  improper  fraction  is  greater  than  1. 
If,  by  taking  just  the  same  number  of  bands  or  rectangles 
as  there  was  in  the  unit,  we  re-form  the  unit,  such  a 
fraction,  consequently,  is  equal  to  1. 

When  we  use  the  word  "  fraction  "  the  word  means, 
as  a  general  rule,  a  proper  fraction.  By  means  of  shading, 
we  can  express  any  fraction  whatever,  leaving  white  those 


FIG.  18. 


FIG.  19. 


parts  which  we  take  from  the  unit,  and  making  dark 
the  bands  or  rectangles  which  are  left.  Thus  in  Fig.  16, 
we  see  the  fraction  three-fifths  ;  in  Fig.  17,  four-fifths ; 
in  Fig.  1 8,  seven-twelfths. 

The  number  of  white  rectangles  (3,  4,  7  in  the  three 
examples)  is  called  the  numerator ;  the  whole  number  of 
rectangles  into  which  the  unit  is  divided  (5,  5,  12)  is 
called  the  denominator ;  and  the  three  fractions  are 

347 

written  thus  -=,  ~  — .     If  we  need  to  represent  an  improper 

O     O     14 

fraction,  we  should  have  no  rectangles  shaded,  and  the 
symbol  would  be  greater  than  the  unit,  the  numerator 
larger  than  the  denominator.  If  the  numerator  is  the 
same  as  the  denominator,  we  have  the  unit  itself. 


Fundamental  Principle. — The  value  of  a  fraction  is  in 

li,  £ 


50  MATHEMATICS 

no  way  changed  by  multiplying  the  numerator  and  denom- 
inator by  the  same  number. 

o 

Let  us  take  the  fraction  -  (Fig.  19).     I  wish  to  show 

TO 

....  15          3  x  5      _,  3 

that  it  is  equal  to  ^Q  or   4  x  «•     The  fraction  7  was 

represented  by  vertical  bands.  I  divide  the  height  into 
5  equal  parts  and  I  picture  the  rectangular  unit  cut  up 
by  horizontal  lines  passing  through  the  points  of  division. 
It  is  thus  divided  into  little  rectangles  all  alike  ;  and  we 
shall  see  that  we  have  now  4  x  5  or  20 ;  then  take  the 

o 

fraction  7 ;    it  contains  3  x  5  or  15  little  rectangles ;    it 

has  not  changed ;    its  denominator  and  its   numerator 

3  3x5 

have  each  been  multiplied  by  5  :    therefore  T  =  -. ^ 

4  4x5 

15 

=  20' 

Two  fractions  can  be  reduced  graphically  to  the  same 
denominator,  either  by  working  on  the  above  principle, 
or  dealing  directly  with  the  figure. 

1  2 

We  will  take,  for  example,  -r  and  =,  the  first  fraction 

represented  by  a  vertical  band,  and  the  other  by  two 
horizontal  bands  (Fig.  20). 

Dividing  the  first  rectangle  into  three  horizontal  bands 
exactly  similar,  the  second  into  four  vertical  bands,  we 

3  8 

see  that  our  two  fractions  read  75  and  ^. 

i  ii          i  — 

Addition  and  subtraction  will  then  be  readily  explained, 
taking  for  illustration  these  concrete  expressions. 

For  multiplication,  the  definition  itself  tells  us  that, 

23  32 

to  multiply  -=  by  7,  we  must  take  j  of  -z. 

2 

Take  (Fig.  21)  the  fraction  v  represented  in  ABCD  by 

two  vertical  bands.     Dividing  the  height  AD  into  four 


THE  DIVIDED  CAKE;   FRACTIONS 


51 


equal    parts,    and    drawing    horizontal    lines,    we    have 

2 
divided  ^  into  four  equal  parts ;    take  three  of   these, 

and  let  us  darken  the  remainder  by  horizontal  shading. 


FIG.  20. 


(3       2\ 
j  of  -  ) ;  it  contains  2x3 
4         O/ 

or  6  little  rectangles  ;  and  the  unit  contains  5  X  4,  or  20. 
Therefore 

2        3  _  2x3        ^ 

5X4~5X4~20' 

It  is  easy  by  such  means  to  formulate  in  the  child's 
mind  the  idea  of  proportion  (the  proportion  of  a  to  & 
being  the  number  which  gives  the  measure  of  a  when  we 
take  b  for  the  unit),  to  show  the 
identity  of  this  proportion  with 


the   fraction  T,  to  establish  that 

a  +  m  .  . 

r— — —   approximates  indefinitely 

to  1  when  we  give  to  m  increasing 
integral  values,  to  make  it  under- 
stood    that     a     fraction     is     the 
quotient  of  the  numerator  by  the  denominator,  to  explain 
clearly  the  fundamental  principles  of  proportions,  etc. 

Whether  with  squared  paper,  or  with  little  squares  or 
rectangles  of  wood,  white  on  one  side  and  black  on  the 
opposite  one,  all  these  operations  can  be  carried  out  in 
the  pupil's  sight.  When  using  materials  such  as  have 

E2 


FIG.  21. 


52  MATHEMATICS 

been  mentioned,  the  work  is  at  once  amusing  and  instruc- 
tive; it  rivets  the  child's  attention,  it  fixes  in  his  mind 
the  essential  truths,  without  which  he  may  be  obliged  to 
force  his  memory ;  he  sees  these  truths,  he  makes  them, 
as  it  were,  with  his  own  hands ;  they  are  no  longer  for  him 
obscure  phrases  repeated  without  any  meaning  attached 
to  them,  but  tangible  realities.  Experience  has  shown 
that  these  methods  are  most  efficacious  from  a  scholastic 
point  of  view  ;  it  is  most  desirable  that  their  use  should  be 
more  and  more  extended. 


22.  We  Start  Geometry. 

We  have  already  seen  what  is  a  straight  line.  It  is 
the  most  simple  of  all  the  geometrical  figures.  We  can 
try  and  extend  our  knowledge  a  little  in  this  respect. 

Let  us  begin,  for  instance,  by 

ft .  ft     forming  an  idea  of    a    plane 

by  looking  at  the  surface  of 

C  • D     a  smooth  piece  of  water,  that 

of  a  good   looking  -  glass,  of 
a  ceiling,  a  floor,  or  a  door. 

A  slate,  a  sheet  of  paper  stretched  on  a  polished  board, 
give  us  also  the  idea  of  a  flat  surface,  and  we  feel  that, 
fike  the  straight  line,  the  plane  may,  in  thought,  be 
prolonged  as  far  as  we  like,  indefinitely.  On  a  flat 
surface  we  can  lay  a  ruler  in  any  direction.  On  a  sheet 
of  paper  we  can  draw  as 
many  lines  as  we  wish.  A  ^^^ 
If  we  draw  two  only, 


they   can    be    (Fig.    22) 

parallel,  as  AB,  CD.     On 

ruled    paper    it    can    be 

plainly    seen    that    the 

lines  on  it  are  all  parallel,  and  that  two  parallel   lines 

never   meet.     If,    on    the   contrary    (Fig.    23),   the   two 

straight  lines  AB,  CD,  meet  at  the  point  O,  the  figures 


WE   START  GEOMETRY  53 

AOC,  COB,  BOD,  DOA  are  angles.  Two  angles  are  equal 
when  we  can  place  one  over  the  other.  The  angles  AOC, 
BOD,  for  example,  are  equal ;  it  is  the  same  with 
COB,  DOA. 

WThen  two  straight  lines  (Fig.  24)  cut  each  other  in  such 
a  way  that  the  angles  DOA,  AOC  are  equal,  the  four  angles 
round  O  are  all  equal ;  we  call  them  then  right  angles, 
and  the  figure  formed  by  the  A 

two  lines  is  that  of  a  cross. 
On  squared  paper,  we  can 
see  right  angles  everywhere 

where  two  lines  meet.    When     Q 

two  straight  lines  form  right 

angles  in  this  way  we  say  that 

they  are  perpendicular  to  one  B 

another.  FI(J  ^ 

An  angle  less  than  a  right 

angle,  like  AOC,  in  Fig.  23,  is  called  an  acute  angle ;  if 
greater  than  a  right  angle,  like  COB,  it  is  an  obtuse 
angle.  A  plumb-line  shows  a  straight  line  which  is 
called  vertical.  A  straight  line  perpendicular  to  a  vertical 
line  is  called  horizontal.  All  straight  lines  drawn  on  the 

surface     of    smooth    water 

A B      would  be  horizontal   straight 

^ rv      lines,  and  this  surface  is  in 

^  ~"  itself  a  plane,  which  we  call 

£ F      horizontal.      On    a    piece    of 

FlG  2-  ruled   paper  laid    before    us, 

the  lines  which  run  from  left 

to  right  are  called  horizontal,  and  the  others  vertical, 
because  we  suppose  that  the  paper  is  lifted  up  and  laid 
against  a  wall. 

Let  us  imagine  that  on  a  sheet  of  paper  we  have  drawn 
three  straight  lines.  These  may  be  considered  in  several 
ways.  The  three  straight  lines  (Fig.  25)  can  be  parallel. 
In  the  second  place  (Fig.  26)  two,  AB,  CD,  might  be 
parallel,  and  the  third,  EF,  might  intersect  them  at  E,  F. 


54  MATHEMATICS 

This  third  line  is  called  a  secant.  In  this  figure  all  the 
angles  marked  1  are  equal  to  each  other;  the  angles 
marked  2  are  also  equal  to  each  other  ;  and  the  sum  of  an 
angle  1  and  of  an  angle  2  is  equal  to  two  right  angles. 
It  might  happen  (Fig.  27)  that  our  three  straight  lines 


A m — a 


c~7i — ° 

FIG.  26.  FIG.  27, 

passed  through  the  same  point  O  ;    we  would  say  then 
that  they  were  concurrent. 

Finally  (Fig.  28),  f  none  of  the  preceding  circumstances 
occur,  the  three  straight  lines  will  cross  each  other  twice  at 
the  three  points  A,  B,  C,  and  will  limit  a  portion  of  the  plane 
ABC,  which  we  can  consider  apart  (Fig.  29),  and  which 
is  called  a  triangle.  The  points  A,  B,  C  are  called  the 
apexes,  and  the  segments,  AB,  BC,  CA  the  sides,  of  the 
triangle.  We  say  that  the  angles  A,  B,  C  marked  on 

the   figure    are    the  angles 
of  the  triangle. 

One  of  the  angles  of 
the  triangle  can  be  a 
right  angle ;  we  say  then 

/"x.          that  the  triangle    is    right- 
^     angled.       It     may     also 
(Fig.  31)   have    an    obtuse 
angle  ;    we    say   then   that 
the  triangle  is  obtuse-angled. 

If  a  triangle  like  those  of  Fig.  32  has  two  sides  AB,  AC 
which  are  equal,  the  triangle  is  termed  an  isosceles  triangle. 
The  angles  B  and  C  are  then  equal. 

If  a  triangle  has  its  three  sides  equal,  it  is  equilateral. 
Its  three  angles  are  then  equal  (Fig.  33). 


55 

In  a  triangle  ABC  (Fig.  34),  we  can  choose  any  side, 
BC,  and  call  it  the  base.  If  we  then  draw  from  the 
point  A  a  straight  line  perpendicular  to  BC,  and  which 


B  C  A  8 

FIG.  29.  FIG.  30. 

meets  BC  at  A',  we  say  that  AA'  is  the  height  or  altitude 
of  the  triangle. 
This    simple    figure,    the    triangle,    has    innumerable 


B 


FIG.  31. 


properties ;  we  will  examine  some  of  them  later.  At 
the  moment  we  must  not  deceive  ourselves :  we  are 
learning  nothing  at  all;  we  are  simply  looking  at  the 


FIG.  32. 


figures  and    getting    to    know    their    names.      That    is 
something  useful,  at  any  rate. 

When  a  part  of  a  plane  (Fig.  35)  is  limited  by  several 
straight  lines,  or  rather  by  several  segments  of  straight 
lines,  this  figure  is  called  a  polygon.  The  segments  AB, 


56 


MATHEMATICS 


BC,  .  .  .  HA  are  the  sides,  the  points  A,  B,  .  .  .  H,  the 
corners  or  apexes,  the  angles  marked  A,  B,  .  .  .  H  the 
angles  of  the  polygon. 

A  polygon  like  that  of  Fig.  36  is  said  to  have  re-entering 


B      A' 


FIG.  33. 


FIG.  34. 


B 


H 


angles.  When  there  is  no  re-entering  angle,  as  in  Fig.  35, 
the  polygon  is  convex.  Generally  speaking,  we  shall  only 
deal  witn  convex  polygons.  A  straight  line  like  AD 
(Fig.  35),  which  joins  two  corners  of  a  polygon,  and  which 

is     not     a     side,     is    called    a 
diagonal. 

In  a  polygon,  the  number  of 
the  corners,  the  sides,  and  the 
angles  are  the  same.  Special 
names  have  been  given  to 
different  polygons,  according 
to  the  number  of  sides  which 
they  possess.  To  begin  with, 
E  as  we  have  already  said,  a 

FIG.  35.  polygon  with  three   sides  is  a 

triangle.     Then 

A  polygon  of  4  sides  is  a  quadrilateral 
„         „        5         „         pentagon 
hexagon 
heptagon 
octagon 
decagon 


6 

' 

10 
12 


dodecagon. 


Fio.  36. 


WE  START  GEOMETRY  57 

Thus,  Fig.  35  represents  a  convex  octagon,  and  Fig.  36 
is  a  heptagon  with  re-entering  angles.  In  a  quadri- 
lateral, the  two  sides  AB,  CD  can  be  parallel  (Fig.  37), 
the  two  others  not  being  so;  such  quadrilaterals  are 

A         B 


Fro.  37. 

called  trapeziums.  The  sides  AB,  CD  are  the  bases  of 
the  trapezium. 

If  (Fig.  38)  the  sides  AB,  CD  are  parallel,  and  if  the  sides 
BC,  AD  are  also  parallel,  the  quadrilateral  is  a  parallelo- 
gram. Then  the  sides  AB  and  CD  are  equal,  and  so  are 
the  sides  BC  and  AD.  Also,  the  angles  A,  C  are  equal, 
and  equally  so  the  angles  B,  D. 

If  in  a  parallelogram  the  four  sides  are  equal,  it  is  a 
rhombus  (Fig.  39). 

If  (Fig.  40)  one  of  the  angles  is  a  right-angle,  the  three 


OH  C 

Fia.  38. 

others  are  right-angles  also,  and  the  parallelogram  is  a 
rectangle. 

If,  finally  (Fig.  41),  a  rectangle  has  all  its  sides  equal,  it 
is  called  a  square. 

In  every  quadrilateral  there  are  two  diagonals ;  in 
every  parallelogram  (Figs.  38,  39,  40,  41)  these  two 


58 


MATHEMATICS 


FIG.  40. 


diagonals  AC,  BD  cut  each  other  at  a  point  O,  which  is 
the  middle  of  each  of  them.  In  a  rhombus  (Fig.  39) 
the  two  diagonals  are  perpendicular  to  one  another. 
T>  In  a  rectangle  (Fig.  40)  the  two 
diagonals  are  equal.  In  a  square 
(Fig.  41)  the  two  diagonals  are  both 
equal  and  perpendicular.  It  will  be 
C  noticed  that  a  square  is  both  a  rhom- 
bus and  a  rectangle. 

If  in  the  parallelogram  (Fig.  38) 
we  take  a  side  CD,  which  we  call  the  base,  and  if  we 
have  a  straight  line  AH  perpendicular  to  CD  (and  also 
perpendicular  to  AB),  this  straight  line,  or 
rather  this  segment  AH,  is  termed  the 
height  of  the  parallelogram. 

On  squared  paper,  by  following  the 
squared  lines,  we  can  form  as  many 
rectangles  and  squares  as  we  like.  The 
various  figures  of  which  we  have  spoken, 
and  others  which  we  can  imagine,  should 
be  constructed  time  after  time  by  the 
pupil,  with  the  help  of  a  pencil,  a  ruler,  a  set  square, 
and  a  measure  to  measure  the  lengths.  Every  line  should 

be  drawn  with  the  greatest 
possible  care.  Afterwards 
he  must  accustom  himself 
to  draw  them  correctly 
without  the  help  of  any 
instrument.  For  this  pur- 
pose it  will  be  well  for 
him,  after  having  drawn  his 
figure  in  pencil  with  his 
instruments,  to  draw  it 
freehand  afterwards  in  ink. 
We  are  not  saying  any- 
thing yet  about  the  use  of  the  compass,  and  the  pro- 
tractor ;  we  shall  touch  briefly  upon  this  later  on. 


WE  START  GEOMETRY 


59 


FIG.  43. 


Moreover,  do  not  let  us  lose  sight  of  the  fact  that  the 
child  must  never  have  left  off  drawing  since  he  began  to 
trace  his  first  strokes. 

When  (Fig.  42)  we  have  a  polygon,  ABCDE,  on  a  hori- 
zontal plane,  for  instance,  if  we  draw  the  straight  lines 
AA',  BB',  CC',  DD',  EE',  all 
parallel  and  equal  to  one  another, 
outside  the  plane,  the  extremi- 
ties, A',  B',  C',  D',  E',  are  the 
corners  of  another  polygon  like  the 
first.  The  quadrilaterals  AA',  BB' 
.  .  .  are  then  parallelograms;  the 
space  which  would  be  limited 
by  all  these  parallelograms  and 
by  the  two  polygons  is  called  a 
prism  ;  the  two  polygons  are  the 
bases;  the  parallelograms  are  the 
faces  ;  the  distance  between  the  planes  of  the  two  bases 
is  called  the  lieight.  The  straight  lines  A  A',  BB'  .  .  . 
are  the  edges. 

If  the  edges  are  vertical  (supposing  the  bases  to  be 
horizontal)  the  prism  is  right-angled. 

If  the  bases  are  parallelograms, 
the  prism  is  called  a  parallelo- 
piped. 

If,  finally,  the  base  is  a  square, 
and  if  the  parallelepiped,  being  right- 
angled,  has  for  its  height  the  side  of 
the  base,  the  parallelepiped  then 
takes  the  form  of  one  of  a  set  of 
dice,  and  is  called  a  cube  (Fig.  43). 
When  (Fig.  44)  we  have  a  polygon, 
ABCDE,  if  we  join  all  the  corners  with  a  point  S  outside 
the  plane,  the  space  which  would  be  limited  by  the 
polygon  and  the  triangles  SAB,  SBC,  .  .  .  SEA,  is  called 
a  pyramid;  ABCDE  is  the  base;  the  triangles  SAB 
.  .  .  are  the  faces  ;  SA,  SB,  .  .  .  are  edges  ;  S  is  the  apex ; 


FIG.  44. 


60  MATHEMATICS 

the  distance  from  the  apex  to  the  plane  of  the  base, 
which  would  be  vertical  if  the  base  was  horizontal,  is 
the  height  of  the  pyramid. 

With  some  small  sticks  and  bits  of  wire  it  is  easy  to 
learn  to  construct  little  models  giving  a  sufficiently  exact 
idea  of  the  figures  we  have  just  mentioned.  We  can  also 
cut  them  with  a  knife  out  of  a  carrot,  or  a  potato 

Notice  that  the  Figs.  42  and  44,  in  perspective,  are 
made  assuming  all  the  edges  to  be  visible  (figures  in 
little  sticks),  whilst  in  the  cube  of  Fig.  43  we  find  three 
unseen  edges  AD,  DC,  DD'  (indicated  by  dotted  lines), 
which  takes  place  if  the  cube  is  a  solid  body. 


23.  Areas. 

The  word  "  geometry  "  signifies,  by  its  etymology,  the 
measurement  of  the  earth.  That  hardly  answers  to 
geometrical  science  such  as  we  know  it  to-day,  but  it 
throws  a  light  upon  the  origin  of  this  science,  which  has 
arisen,  like  others,  from  the  needs  of  the  human  race. 
From  quite  early  times  men  have  recognised  the  need 
for  estimating  the  extent  of  pieces  of  land,  and  have  sought 
the  best  means  of  arriving  at  it.  These  pieces  of  land 
being  pretty  nearly  flat  on  the  whole  and  more  often  than 
not  limited  by  straight  lines,  it  follows  that,  to  acquaint 
ourselves  with  the  extent  of  land,  we  must  determine 
and  estimate  the  extent  of  the  various  polygons  described 
in  the  preceding  chapter. 

But  to  measure  anything,  no  matter  what,  we  must 
have  a  unit.  We  know  how  to  measure  lengths,  taking 
for  a  unit  a  metre,  or  a  match,  or  the  side  of  a  division 
of  squared  paper.  To  measure  length  we  must  have  a 
unit  of  length.  To  measure  a  flat  expanse,  wliich  is 
called  an  area,  we  must  start  with  a  unit  which  is  in  itself 
an  area. 


AREAS 


61 


FIG.  45. 


Invariably,  the  unit  of  length  having  been  chosen, 
the  area  unit  will  be  the  area  of  the  square  having  for 
its  side  this  unit  of  length. 

If,  to  measure  a  length,  it  suffices  to  lay  out  the  chosen 
unit  end  to  end,  and  to  count  the  number  of  times  that 
we  have  thus  laid  it  out,  it  is  easy  to  understand  that  such 
a  proceeding  is  practically  impossible  when  dealing  with 
an  area ;  the  squared  unit  must 
be  laid  out  so  that  the  area  will 
be  covered,  which  cannot  be  done. 

On  the  other  hand,  for  the 
figures  described  further  back, 
there  are  very  simple  ways  of 
determining  their  areas. 

To  begin  with,  let  us  take  a 
square.  We  will  take  squared 
paper,  of  which  we  suppose  that 
each  division  is  the  unit  of  length. 
Consequently  each  square  will  be  the  area  unit. 

On  this  squared  paper  we  will  draw  (Fig.  45)  a  square 
whose  side  will  contain  7  divisions,  so  that  the  length 
of  this  side  has  7  for  its  measure.  The  squares  contained 
in  this  figure  are  made  up  of  7  rows,  each  containing  7 
squares  ;  their  total  number  then  is  7  times  7,  or  7  x  7  = 

72  =  49.  And  using  a  to  indi- 
cate the  number  of  divisions 
on  the  side  of  a  square  (whether 
this  number  be  7  or  any  other 
number),  the  area  of  the  square 
would  be  a  x  a  =  a2 ;  that  is  to 
say,  the  number  which  measures  the  area  of  the  square 
is  the  2nd  power  of  the  number  which  the  side  measures. 
It  is  for  this  reason  that  the  square  of  a  number  is 
called  its  2nd  power. 

We  will  take  now  (Fig.  46)  a  rectangle  whose  sides  are 
8  and  3  ;  the  number  of  squares,  that  is  to  say,  the  number 
which  will  measure  its  area,  will  be  8  x  3 ;  if,  instead 


'  FIG.  46. 


62 


MATHEMATICS 


of  8  and  3,  we  have  a  and  b,  the  area  of  the  rectangle  will 
be  measured  by  the  product  ab. 

Now  imagine  (Fig.  47)  a  parallelogram  ABCD,  and  take 
its  height  CH.  If,  having  formed  this  parallelogram  in 
cardboard,  for  instance,  we  cut  off,  by  a  line  along  CH, 
the  triangle  CHB  which  is  shaded  on  the  figure,  and  if 
we  set  down  this  triangle  on  the  left,  laying  CB  on  DA,  we 
form  the  rectangle  CDKH,  whose  area  will  be  the  same  as 
that  of  the  parallelogram,  since  it  is  made  up  of  the  same 
pieces.  This  rectangle  has  for  its  sides  the  base  CD  of 
the  parallelogram,  and  its  height  CH.  Then  the  area  of 
a  parallelogram  has  for  its  measure  the  product  of  the 
numbers  which  measure  its  base  and  its  height. 


H 


•B 


B 


FIG.  47. 


FIG.  48. 


A  parallelogram  (Fig.  48)  being  cut  in  two  by  a  line 
along  the  diagonal  AC,  the  two  triangles  CBA,  ADC  will 
lie  exactly  one  upon  the  other.  Then  the  parallelogram  has 
an  area  double  that  of  the  triangle  ADC,  and  the  latter 
has  an  area  half  of  that  of  the  parallelogram.  Making 
the  product  of  the  base  DC,  by  the  height  AH,  and  taking 
the  half  of  this  product,  we  shall  have  the  number  measur- 
ing the  area  of  the  triangle. 

A  trapezium  (Fig.  49)  can  be  split  up  in  the  same  way 
into  two  triangles.  We  deduce  from  this  that  to  obtain  the 
number  which  measures  its  area,  it  is  necessary  to  multiply 
the  height  AH  by  the  half  of  the  sum  of  its  bases  AB,  DC. 

We  can  also  (Fig.  50,  A)  transform  a  trapezium  into  a 
rectangle  of  the  same  area,  It  is  only  necessary  to  draw 
the  heights  HK,  IJ  through  the  centres  L,  M,  of  the  sides 
AD,  BC.  The  triangular  shaded  portions  LDH,  MCI, 


AREAS 


63 


can  be  laid  exactly  on  LAK  and  MBJ,  and  thus  form  the 
rectangle.  This  shows  us  that  HJ,  or  LM,  is  equal  to  half 
of  the  sum  of  the  bases  AB,  CD. 

Finally  (Fig.  50,  B),  if  we 
prolong  the  side  AB  by  a 
Irngth  BF,  equal  to  DC,  and 
the  side  DC  by  a  length  CG, 
equal  to  AB,  the  figure  AFGD 
is  a  parallelogram;  and  cutting 
it  along  BC  we  have  two 


DH 


FIG.  49. 

trapeziums,  which  will   fit  over  one  another ;    each  of 
them  is  then  the  half  of  the  parallelogram,  which  gives 

R  A  B       0  A 


D  H  100 

(A)  (B) 

FIG.  50. 

us  yet  again  the  area  of  a  trapezium  by  a  new  means, 
merely  by  a  simple  cut  of  the  scissors  across  the  card- 
board parallelogram. 

We  can  summarise  what  has  gone  before  in  the  following 
formulae : — 
Square  Side  A 

Rectangle  Sides  A,  B 

Parallelogram     Base  A  ;  height  H 

Base  A,  height  H 


Area  A2 
Area  AB 
Area  AH 
AH 


Triangle  Base  A,  height  H  Area 

Trapezium  Bases  A,  B,  height  H     Area 


(A  +  B)  H 


Moreover,  as  soon  as  the  pupil  can  measure  the  area  of 
a  triangle,  he  can  determine  that  of  any  polygon  whatever, 
ABCDEF  (Fig.  51),  since  by  the  diagonals  AC,  AD,  AE, 
starting  from  a  corner,  he  can  cut  the  polygon  into  the 
triangles  ABC,  ACD,  ADE,  and  AEF- 


64  MATHEMATICS 

To  determine  in  this  manner  exact  areas,  such  as  those 
of  a  door,  a  window,  a  table,  the  floor  or  the  ceiling  of  a 
room,  a  playground,  etc.,  much  practice  will  be  necessary. 
A  tape  measure  will  be  sufficient  to  use  for  this  purpose. 
According  to  the  object,  he  will  take 
for  the  unit  of  length  the  yard,  the 
foot,  the  inch,  the  metre,  the  deci- 

«  /^ \       metre  and  the  centimetre  ;  without 

X^^^^    "/        having  up   to   this   point  any  idea 
N^J^^o         of  theory,  in   this   manner   he  will 
g  familiarise   himself   with    the    most 

simple  applications  of  our   weights 
FIG.  51.  an(j    measures   and    of    the    metric 

system ;  he  will  grasp  them  intuitively ;  he  will  have  an 
exact  idea  of  the  employment  of  the  various  units  ;  and 
this  acquisition,  already  useful  hi  itself,  will  become  later 
a  very  considerable  help  when  he  really  begins  his  studies. 

24.  The  Asses'  Bridge. 

There  is  in  geometry  a  proposition  at  once  celebrated 
and  important,  but  which  has  been  the  despair  of  many 
generations  of  scholars,  because  the  academical  demonstra- 
tion that  is  usually  given  of  it  is  hardly  natural,  and 
difficult  to  remember.  It  is  known  under  different 
names ;  it  is  called  "  the  square  of  the  hypotenuse," 
"  the  theorem  of  Pythagoras  "  (although  it  was  known 
many  centuries  before  Pythagoras),  lastly  "  the  asses' 
bridge,"  undoubtedly  because  ordinary  scholars  stumble 
at  it  and  have  some  trouble  in  getting  over  it. 

We  already  know  what  a  rectangular  triangle  is.  The 
greatest  side  BC  (Fig.  52),  that  which  is  opposed  to  the' 
right  angle,  is  called  the  hypotenuse.  If  three  squares  be 
formed,  BDEC,  CFGA,  AHIB,  having  for  their  sides  the 
hypotenuse  and  the  two  other  sides,  the  area  of  the  first 
will  be  equal  to  the  sum  of  the  areas  of  the  two  others. 
This  is  the  opening  statement  of  the  famous  asses'  bridge. 


THE  ASSES'  BRIDGE 


65 


H 


Now,  there  is  a  very  simple  method  of  verifying  this 
proposition,  a  method  which  was  invented  in  India  in 
the  very  earliest  times  and  can  be  used  for  a  most  exact 
demonstration  when  the  study 
of  geometry  has  been  begun 
—though  as  yet  we  have  not 
entered  on  it. 

Let  us  take  a  square  (Fig. 
53  (1)  )  whose  side  is  AB. 
Marking  a  point  C  between  A 
and  B,  we  will  construct, 
either  in  wood  or  cardboard, 
rectangular  triangles  having 
AC  and  CB  for  their  sides 
which  contain  the  right  angle. 
Four  will  be  sufficient.  Arrange 
them,  numbering  them  1,  2, 
3,  4,  as  they  are  shown  in 


FIG.  52. 


Fig.  53  (1),  where  the  shaded  parts  represent  these 
little  triangles.  We  see  that  they  form  a  pattern 
which  allows  a  square  to  be  seen  in  the  interior, 
which  has  for  its  side  exactly  the  hypotenuse.  This 
square  is  then  what  remains  when  a  part  of  the  large 

f2) 


B 


B 


FIG.  53. 

square  has  been  covered  with  the  four  triangles.  Now, 
let  us  slip  our  four  triangles  into  the  position  indicated 
by  Fig.  53  (2).  What  now  remains  is  two  squares,  the 
two  squares  constructed  on  the  sides  of  the  right  angle. 


66 


MATHEMATICS 


Then  both  of  them  have  the  same  area  as  the  square  of 
the  hypotenuse  in  Fig.  53  (1).  It  is  just  a  very  simple 
game  of  patience  ;  the  child  who  has  practised  it  once  or 
twice  will  never  forget  it  in  his  whole  life,  and  will  never 
be  dismayed  nor  confused  when  he  approaches  the  asses' 
bridge.  The  greatest  blunder  of  all  is  to  complicate 
simple  things  and  to  make  difficult  anything  which  is 
easy. 

25.  Yarious  Puzzles ;  Mathematical  Medley. 

On  a  segment  ABC  (Fig.  54)  let  us  construct  a  square 
ACIG  ;  then  taking  CF  -  BC,  let  us  draw  FED  parallel 
to  AC  ;  also  draw  BEH  parallel  to  CI.  The  large  square 
will  be  cut  into  four  parts  by  the  lines  BH  and  FD  ;  this 
can  be  done  by  two  snips  of  the  scissors.  These  four 
pieces  are  : 

1st  BCFE,  square  having  for  its  side  BC. 

2nd  EHGD,  square  having  for  its  side  DE  which  is 

equal  to  AB. 
3rd  EFIH,  rectangle  having  its  sides  equal  to  AB, 

BC. 

4th  ABED,  rectangle  like  the  preceding  one. 
We  have  just  verified  this  theorem  of  geometry  : 

"  The  square  constructed   upon 

6 H          I       the  sum  of  two  lines   is  equal  to 

the  square  constructed  upon  the 
first,  plus  the  square  constructed 
upon  the  second,  plus  twice  the 
rectangle  constructed  with  these 
two  lines  as  sides." 

If  we  have  drawn  the  figure 
on  squared  paper,  by  estimating 
the  areas  of  all  these  figures,  that 
is  to  say,  counting  the  divisions, 


D 


E 


A  a 

FTG.  54. 

we  have  the  proposition  of  arithmetic  : 

"  The  square  of  the  sum  of  two  numbers  is  equal  to 


VARIOUS  PUZZLES 


67 


the  sum  of  the  squares  of  these  two  numbers,  plus  twice 
their  product." 

If  we  indicate  AB  by  a,  BC  by  b,  we  have  finally  this 
formula  in  Algebra : 

(a  +  6)2  =  a2  +  2ab  +  62. 

These  are  three  truths  which  are  loaded  on  the  memory 
of  the  unprepared  child  three 
times,  while  in  reality  they  are 
all  one  thing  which  jumps  to  the 
eye.  Its  appearance,  its  dress, 
is  different,  but  it  is  itself  the 
same  in  each  case.  Knowing  this 
beforehand,  he  will  be  spared 
loss  of  time  and  vain  efforts, 
and,  more  than  all,  will  know 
that  these  classifications  are 


G      fl 


t       J 


D      E 


A  B 

FIG.  55. 


H      I 


J 


F     G 


necessary,  but  often  artificial  by  the    force    of   things ; 

and  will  accustom  himself  early  to  recognise  the  analogies 

he  will  encounter. 

We  are  going  to  mention  others.     Let  us  form  (Fig.  55) 

on  the  segment  ABC  a  square  having  for  one  of  its  sides 

AB,  which  square  will  be  ABFE, 
and  a  square  ACJH,  having  a 
side  AC.  Produce  BF  to  I, 
and  on  EH  construct  the  square 
EHGD. 

For  the  square  ABFE,  we 
must  take  away  from  the  whole 
figure  the  rectangles  BCH, 
FIGD ;  the  whole  figure  is  made 
by  the  reunion  of  two  squares 

whose  sides  are  equal  to  AC  and  BC  ;  the  two  rectangles 

are  similar,   and  their  sides  are  equal  to  AC   and   BC ; 

finally  AB  is  the  difference  of  AC  and  BC.     Then  : — 
Geometry. — The  square  constructed  on  the  difference 

of  two  segments  is  equal  to  the  sum  of  the  squares  con- 

F2 


FIG.  56. 


68  MATHEMATICS 

structed  on  these  two  segments,  less  twice  the  rectangle 
constructed  with  the  two  segments  as  sides. 

Arithmetic. — The  square  of  the  difference  of  two  numbers 
is  equal  to  the  sum  of  the  squares  of  these  numbers,  less 
twice  their  product. 

Algebra.— This  formula  will  be  (a  -  6)2  =  a2  -  2ab  +  62. 

One  more  example  (Fig.  56) ;  ABJH  is  a  square,  ACGD 
a  rectangle ;  FG,  FJ,  DE  are  equal  to  BC,  DEIH  is  a 
square. 

The  rectangle  ACGD  has  thus  for  sides  AB  +  BC  and 
AB  -  BC  ;  as  the  two  rectangles  BCGF,  FJIE  are 
identical,  by  taking  away  the  first  and  putting  it  in  the 
place  of  the  second  we  shall  have  ABJIED,  which  is 
the  difference  of  the  squares  ABJH,  DEIH  constructed 
on  AB  and  DE  =  BC.  Therefore  :— 

Geometry. — The  rectangle  having  as  sides  the  sum  and 
the  difference  of  two  segments  is  equal  to  the  difference 
of  the  squares  having  these  two  segments  as  sides. 

Arithmetic. — The  product  of  the  sum  of  two  numbers 
by  their  difference  is  equal  to  the  difference  of  their 
squares. 

Algebra. — This  formula  will  be  (a  +  b)  (a  —  b)  =  a?  —  b2. 

And  to  verify  so  many  propositions,  concerning  so 
many  sciences,  it  will  only  be  necessary  to  cut  some 
shapes  of  cardboard  into  pieces,  after  having  made  the 
figures  with  great  care. 

These  games  of  cutting  up  cardboard  have  sometimes 
been  called  brain  puzzles.  This  is  very  unjust,  because 
used  in  the  manner  we  have  just  indicated  they  prevent 
on  the  contrary  much  puzzling  of  the  brain  in  the  future 
by  dint  of  teaching  by  means  of  the  eye. 

26.  The  Cube  in  Eight  Pieces. 

Let  us  take  (Fig.  57)  a  wooden  cube,  and  starting 
from  one  of  the  corners  O,  let  us  lay  out,  on  the  three 
edges  which  end  there,  three  lengths  equal  to  each  other, 


THE  CUBE   IN  EIGHT  PIECES 


69 


OA,  OB,  OC.     Suppose  that  we  saw  through  along  AAA, 
BBB,  CCC,  at  each  of  the  three  points  thus  obtained. 

By  this  means  the  cube  is  cut 
into   eight   pieces.      To  make   the  A 

explanation  easier,  by  means  of 
looking  at  the  object  itself,  let  us 
call  (Fig.  57)  the  length  DA  a,  D 
and  the  length  AO  b,  constructing  £ 
thus  Fig.  58.  The  two  parts  of 
which  it  is  composed  represent 
what  we  see  after  the  cuts  along 
AAA,  and  BBB,  when  we  look 
at  the  cube  from  above.  As  well 
as  this,  the  letters  (a)  (b)  between 
brackets  show  the  thickness  after  the  cut  along  CCC. 
The  left  figure  shows  what  is  underneath  CCC,  and  that 
on  the  right  what  is  above. 


/       / 

Y 

/               / 

/ 

A 

0 

C 

A       B 

FIG.  57. 

tai 

(3) 

(3) 

(31 

a         b 

FIG.  58. 

We  can  easily  see  that  we  shall  have  eight  parallele- 
pipeds whose  dimensions  will  be  : 

Figure  on  the  left,     aaa,   aba,   bba,   baa. 
Figure  on  the  right,   aab,    abb,   bbb,    bob. 

That  gives  us  then  : 

a  cube  whose  edge  is  a  ; 


3  parallelepipeds  having  for  sides  a,  a,  b  ; 
3  „  „  „        a,  b,  b  ; 


70  MATHEMATICS 

The  edge  of  the  cube  which  we  have  cut  into  eight 
pieces  was  a  +  b. 

We  verify  thus  that  the  cube  constructed  upon  the  sum 
of  two  segments  a,  b  is  made  up  : — 

First,  of  the  sum  of  the  cubes  constructed  upon  each 
of  the  segments ; 

Second,  of  three  times  a  parallelepiped,  having  for  its 
base  a  square  with  the  side  a,  and  for  its  height  b  ; 

Third,  of  three  times  a  parallelepiped,  having  for  its  base 
a  square  with  a  side  b,  and  for  its  height  a. 

This  is  Geometry. 

The  same  figure  shows  us  that  in  Arithmetic  the 
cube  of  the  sum  of  two  numbers  is  equal  to  the  sum  of 
the  cubes  of  these  two  numbers,  plus  three  times  the 
product  of  the  first  by  the  square  of  the  second,  plus 
three  times  the  product  of  the  second  by  the  square  of 
the  first. 

Finally  (Algebra)  this  gives  the  formula  : 

(a  +  by  =  a3  +  3a26  +  3a62  +  63. 

This  is  quite  analogous  to  what  we  have  done  for  the 
square  of  a  sum  in  the  preceding  section. 

With  a  sufficient  number  of  little  wooden  cubes,  the 
constructions  which  we  have  indicated  may  be  made,» 
and  also  many  more.  These  are  games  which,  directed 
with  a  little  method,  help  the  child  very  much  to  see  the 
figures  in  space,  and  engage  his  attention. 

If  necessary,  the  cutting  up  of  the  cube  might  be  done 
by  means  of  a  piece  of  soap  taken  from  a  bar,  cutting 
it  carefully  with  a  wire  instead  of  using  a  saw.  But  the 
wooden  cube  is  much  to  be  preferred,  and  is  certainly 
neither  difficult  nor  expensive  to  procure. 

27.  Triangular  Numbers. — The  Flight  of  the  Cranes. 

Edward  Lucas  attributes  the  origin  of  the  numbers 
which  have  been  called  triangular  to  the  observation  of 
the  flight  of  certain  birds.  At  the  head  there  flies  a 


TRIANGULAR  NUMBERS 


71 


FIG.  59. 


single  bird  ;   behind  him,  on  a  second  line,  there  are  two  ; 
on  a  third  line  behind  them,  there  are  three,  and  so  on  ; 
so   that   the   general   disposition   of   the   flying   column 
presents    the     appearance     of     a 
triangle. 

It  is  easy  to  give  ourselves  a 
precise  idea  of  these  numbers 
and  to  represent  them  on  a 
squared  pattern ;  looking  at  Fig.  59, 
for  example,  and  considering,  to 
begin  with,  the  part  A  only, 
which  shows  us,  on  the  top,  one 
division,  then  two  divisions  in 
a  second  row,  then  3,  4,  5,  6,  7 
divisions  in  the  following  rows,  up  to  the  seventh. 

We  have  then  the  7th  triangular  number 

1+2  +  3  +  4  +  5  +  6  +  7; 

to  find  its  value,  we  can  add  up,  which  will  give  us  28. 
But  that  would  teach  us  nothing  about  any  other  tri- 
angular number.  If  we  wanted  to  have  the  1  000th,  for 
instance,  we  would  have  to  add  from  1  to  1,000,  which 
would  be  long  and  very  wearisome.  Instead  of  that, 
let  us  now  look  at  the  whole  of  Fig.  59  ;  the  part  B,  if 
we  look  at  it  from  the  bottom  to  the  top,  or  if  we  turn 
it  upside  down,  shows  us  still,  by  the  number  of  its 
squares,  the  same  triangular  number.  The  entire  figure 
then  represents  twice  the  triangular  number  in  question  ; 
and  as  it  is  composed  of  seven  rows,  each  having  eight 
divisions,  the  total  number  of  divisions  is  7  x  8,  and  the 
required  number  will  be  the  half  of  this  product,  that 
is  to  say,  28. 

We  shall  have,  in  other  terms, 

1+2  +  3  +  4  +  5  +  6  +  7=™  =  28. 

If  we  wanted  to  get  the  1,000th  triangular  number, 
supposing  that  we  did  it  the  same  way,  we  should  have 


72  MATHEMATICS 

1  +  2  +  3  +  .      .  +  1,000  =  1)0°0:1>001  =  500,500. 

31 

This  is  shorter  than  adding. 

And  as,  instead  of  1,000,  we  might  have  any  whole 
number  n,  we  have  also 


...  , 

which  expression  will  allow  us  to  find  the  nth  triangular 
number,  which  we  can  call  TB. 

The  total  number  of  divisions  in  Fig.  59  is  2  Tn.  If 
we  take  away  the  last  column,  a  square  of  seven  lines 
will  remain,  each  containing  7  divisions.  We  see,  there- 
fore, that  the  new  figure  is  formed  of  the  combination 
of  the  triangular  numbers  T6  and  T7.  We  have  therefore 
2  T7  -  7  =  72  =  T7  -f  T6. 

If  we  add  below  a  new  row  of  eight  divisions,  we  see 
that  we  have 

2  T7  +  8  =  82  =  T8  +  T7, 
just  by  looking  at  the  figure. 

And  as,  in  place  of  7,  we  might  have  taken  any  other 
number  n 

2  Tn  -  n  =  n-  =  Tn  +  Tn_: 

2  Tn  +  n  +  1  =  (n  +  I)2  =  Tn+1  +  Tn. 

These  formulae,  which  appear  very  learned,  do  not, 
however,  require  even  the  least  calculation,  since  the 
pupil  can  read  them  from  figures,  since  he  can  see  them, 
since  he  can  make  them  with  his  hands  by  means  of 
little  wooden  squares,  or  even  with  simple  counters  by 
placing  one  in  each  division. 

28.  Square  Numbers. 

Take  (Fig.  60)  a  square,  composed  of  7  rows,  of  7 
divisions  each,  in  all  7.7=  72  =  49  divisions.  On 
this  figure,  by  means  of  traced  lines,  we  see  the  successive 
squares,  1,  22,  32,  42,  52,  62  divisions. 


SQUARE  NUMBERS 


73 


The  first  square,  of  1  division,  is  represented  by  the 
division  at  the  top  left-hand  side.  To  pass  from  this 
square  to  that  of  22  or  4 
divisions,  we  notice  that  it 
is  necessary  to  add  3  divi- 
sions, so  that  1  +  3  =  22; 
to  pass  to  the  following 
square,  of  9  divisions,  we 
must  add  two  from  the  right, 
two  underneath,  and  one 
from  the  right  below,  which 
makes  5  ;  and  by  continuing 
in  the  same  manner,  we  see 
that 

72  =  1+3  +  5+7+9 


ri°-  CO. 
11  +13. 

Which  means  that  the  square  of  7  is  equal  to  the  sum 
of  the  first  7  uneven  numbers. 

Instead  of  7,  let  us  take  any  whole  number  we  like,  n. 
The  first  uneven  numbers  are  1,  3,  5  ,  ,  .  and  the  nth 


1 

1 

i  •       i 

. 

r 

L 

1 

-  1 

1 

i 

Fro.  61. 

is  2n  —  1.     As  we  have  been  able  to  make  this  figure 
up  to  the  number  n,  we  have 

n2  =  1  +  3  +  5  +  .  .  .   +  2n  -  1, 
and  this  only  expresses  what  we  see  in  Fig.  60. 


71  MATHEMATICS 

We  thus  see  that  there  is  another  way  by  which  we 
can  represent  square  numbers  ;  it  is  shown  in  Fig.  61, 
where  we  see  the  squares  of  1,  of  2,  of  3,  and  4.  With 
little  wooden  squares  it  will  be  easy  to  make  and  also 
change  these  various  figures. 

Without  any  trouble,  we  are  now  going  to  solve  a  very 
much  more  difficult  problem,  that  of  finding  the  sum  of 
the  squares  of  1,  2,  3,  4,  for  instance.  By  making  use  of 
Fig.  60,  and  by  laying  the  squares  of  1,  2,  3,  4  from 
bottom  to  top,  we  have  at  once  Fig.  62,  which  needs 


FIG.  62. 


FIG.  63. 


FIG.  64. 


no  explanation.     Making  use  of  the  various  elements  of 
Fig.  61,  we  see  that  there  are 

4  rows  of  1  division 
3       „        3  divisions 

S  M  O  ,, 


which,  placed  each  under  the  other,  give  Fig.  63. 

Let  us  bring  together  Fig.  64,  Fig.  62,  the  same 
turned  over,  and  also  Fig.  63.  We  obtain  a  rectangle 
which  will  contain  three  times  the  required  number 
of  divisions. 

The  number  of  rows  in  this  rectangle  is 


1  +  2  +  8  +  4,  or  -;  -  =  10. 


SQUARE  NUMBERS  75 

The  number  of  divisions  contained  in  each  row  is,  as 
we  can  see  on  the  first  line, 

4  +  1  +  4,  or  2.4  +  1  =  9. 

The  total  number  of  divisions  then  is  10.9  =  90,  and 
the  required  number  will  be  the  third  of  90,  that  is  to 
say,  30.  We  thus  verify  that 

I2  +  22  +  32  +  42  =  1  +  4  +  9  +  16  =  30. 

But  if,  instead  of  4,  we  had  taken  any  number  whatever, 
n,  and  if  we  had  done  exactly  the  same  thing,  the  rectangle 
of  Fig.  64  would  have 

n(n  +  1).. 
l  +  2  +  3+...-}-w,  or     v  0  —  L  lines, 

m 

and  2n  +  1  columns. 

The  total  number  of  its  divisions  would  be  then 
n(n  +  1)  (2n+  1) 
2  ' 

and  to  have  the  required  number  we  must  take  the  third 
of  it,  which  shows  us  that 


This  is  a  formula  which  candidates  at  the  Polytechnic 
School  cannot  always  prove,  giving  themselves  an  infinite 
amount  of  trouble  and  endless  calculation,  whilst  we 
establish  it  by  amusing  ourselves  with  little  wooden  squares 
like  those  we  have  already  seen. 

This  determination  of  the  sum  of  the  squares  of  the 
first  n  numbers  had  formerly  an  important  practical 
application  in  artillery,  when  spherical  projectiles  were 
in  use  (cannon-balls  or  shells).  Indeed,  these  were 
often  arranged  in  arsenals,  by  forming  a  square  on  the 
ground,  then,  above,  another  smaller  square,  and  so  on 
up  to  the  top,  which  was  made  of  a  single  ball  or  shell. 
This  was  called  a  pile  of  balls  with  a  square  base.  Then, 
in  order  to  count  the  balls  contained  in  a  pile,  it  suffices 
to  count  the  number  n  of  the  balls  on  one  side  of  the  base 


76  MATHEMATICS 

and  to  apply  the  above  formula.     For  example,   if  n 
equals  17,  the  required  sum  is 

17.18.35 


6 


or  1785. 


The  child  could  amuse  himself  by  forming  piles  of 
oranges  in  the  same  way,  provided  that  they  are  about 
the  same  size,  or,  more  simply  perhaps,  by  using  billiard 
balls,  laying  them  on  a  light  bed  of  sand  to  keep  them  from 
rolling  and  so  spoiling  the  erection. 

29.  The  Sum  of  Cubes. 

To  represent  a  number  raised  to  the  cube,  such  as 
23  =  2.2.2,  33  =  3.3.3,  etc.,  it  would  be  convenient 
to  have  a  large  number  of  little  wooden  cubes,  rather  larger 
than  dice,  which  would  serve  both  to  make  the  figures 
of  which  we  have  spoken  above,  and  also  to  perform  the 
various  operations  which  are  to  follow. 

We  need  not  be  quite  so  strict,  however,  for  we  can 
dispense  with  the  cubes,  and  replace  each  unit  by  a  small 
flat  square,  of  wood  or  card- 
board, or  even  by  a  simple 
counter.  It  is  this  last  supposi- 
tion that  we  shall  adopt.  Later, 
when  we  have  seen  how  easy  the 
constructions  are,  they  will  be 

made  all  the  more  readily,  by  means  of  squares  or  cubes ; 
he  who  can  do  the  most  can  certainly  do  the  least. 

Begin  by  seeing  how,  with  our  counters,  we  can  represent 
successive  cubes.  The  cube  of  1  is  1  ;  a  single  counter 
will  represent  it. 

The  cube  of  2  is  2  x  2  x  2,  or  8  ;  therefore  it  will  be 
composed  (Fig.  65)  of  2  squares  of  4  counters  each,  the 
squares  placed  side  by  side,  in  the  first  part  of  the  figure. 
But,  as  in  the  second  part,  these  8  counters  may  be 
arranged  in  another  manner,  by  keeping  the  first  three 


o  I  o 


1HE  SUM  OF  CUBES 


77 


columns,  and  by  placing  the  fourth,  which  has  become 
horizontal,  on  top. 
Let  us  proceed  to  the  cube  of  3,  which  is  3  x  3  x  3, 


0 

0 

o 

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o 

Fro.  66. 


or  27  ;  it  is  represented  (Fig.  66)  by  3  squares  of  9  counters 
each,  side  by  side,  in  the  first  part  of  the  figure. 

The  second  part  is  obtained  by  keeping  the  first  six 


0 

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FIG.  67. 

columns  and  placing  on  top  the  last  three,  which  are  now 
horizontal. 
Finally,  for  the  cube  of  4,  we  will  do  the  same,  by  keeping 


78 


MATHEMATICS 


(Fig.  67)  the  first  10  columns  of  the  first  part  and  placing 

above  them  the  last  six,  horizontally,  to  obtain  the  second 

part  of  the  figure. 

If  we  now  put  together  (Fig.  68)  the  second  parts  of 

Figs.  65,  66,  67,  by  adding  a  counter  to  the  left  at  the 

top,  which  will  represent  the 
cube  of  1,  we  shall  have  the 
sum  of  the  cubes  of  1,  2,  3, 
4  in  the  form  of  a  square,  in 
which  the  number  of  the 
counters  in  a  row  or  a  column 
is  1  +  2  +  3  +  4,  or  10.  The 
sum  of  these  cubes  is  then 
100. 

It  will  be  interesting  to 
take  counters  of  different 
colours  to  represent  each  of 
the  cubes.  The  figure  will 


0  J  O 

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o 

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FIG.  68. 
then  be  so  much  the  more  striking. 

The  method  of  construction  indicated  might  thus  be 
carried  out  as  far  as  the  cube  of  any  number  whatever, 
n,  and  shows  that  the  sum  of  the  cubes  of  tlie  n  first  whole 
numbers  is  equal  to  the  square  of  the  sum  of  these  numbers. 

This  is  expressed  by  the  formula 

I3  +  23  +  33  +  . . .  +  n3  =  (1  +  2  +  3  +  . . .  +  n)2. 
It  is  an  expression  which  can  also  be  written  (T,,)2  or 
'n  (n  +  1)\2 


/n(n+l)y 


By  this  means  we  have  a  result  which  is  much  more 
troublesome  and  difficult  to  obtain  by  calculation. 
Here  we  arrive  at  it  by  a  simple  construction  game T  ! 

1  It  is  perhaps  worth  noticing  that  in  our  table  of  multiplication 
without  figures  (Fig.  12)  we  find  exactly  that  the  successive  cubes 
2s,  3s,  .  .  .  represent  the  number  of  divisions  in  the  spaces  separating 
the  squares  of  1,  1  +  2,  1  +  2  +  3.  .  .  .  Indeed,  strictlv,  we  should  have 
been  able,  with  this  simple  table,  to  demonstrate  all  that  we  have  just 
seen  here. 


THE  POWERS  OF   11  79 


30.  The  Powers  of  11. 

If  we  take  the  number  11,  and  wish  to  form  the  square 
of  it,  the  multiplication  will  be  very  easy 

11 
11 

11 
11 


121          II2  =  121. 
To  obtain  the  cube  we  shall  have 

121 
11 

121 
121 


1331          II3  =  1331. 

The  fourth  power  would  need  the  multiplication  as 
below 

1331 
11 


1331 
1331 

14641         II4  =  14641 
Let  us  fix  our  attention  upon  these  figures 

1,  2,  1  ;         1,  3,  3,  1  ;         1,  4,  6,  4,  1, 

which  are  employed  to  write  the  powers. 

We  would  have  been  able  to  have  them,  with  less  writing, 
without  putting  down  the  multiplications,  if  we  notice 
first  that  we  begin  and  end  with  1  ;  and  also  that  we 


80 


MATHEMATICS 


have  only  to  add  two  figures  which  follow  each  other 
to  get  a  figure  of  the  following  power. 

Thus  from  11  we  get  121,  because  1  +  1=2; 

From  121  we  get  1331,  because  1+2  =  3,  2  +  1=3; 

From  1331  we  get  14641,  because  1+3  =  4,  3  +  3  =  6, 
3  +  1=4. 

These  remarks  have  led  to  a  means  of  obtaining  these 
figures  (and  many  other  numbers)  very  easily,  as  we  are 
going  to  see  in  the  following  section. 

It  is  all  the  more  useful  because  the  numbers  of  which 
it  treats  are  of  great  importance  in  algebra,  where  the 
pupil  will  have  to  deal  with  them  later,  however  little 
he  may  study  mathematics. 


31.  The  Arithmetical  Triangle  and  Square. 

Let  us  write  (Fig.  69)  the  figure  1  as  many  times  as  we 
like  each  under  the  other.  Suppose  that  at  the  right  of  the 
first  one,  on  top,  there  are  noughts,  which  it  is  not  neces- 
sary to  write.  We  form  the  second 
line  adding  1  and  0,  which  makes  1, 
and  writing  this  1  to  the  right  of 
that  which  is  already  put  down.  Let 
us  pass  on  to  the  third  line  ;  we  read 
in  the  second,  1  and  1  making  2,  which 
we  write  ;  then  1  and  0,  1,  which  we 
place  to  the  right  of  the  2  ; 
similarly,  starting  from  the  3rd  line 

we  form  the  4th,  1  and  2,  3  ;    2  and 
r  IG.  69.  ,  , 

1,    3 ;    1    and   0,    1.      And    so   on    as 

far  as  we  like.      The  first  rows  of  the  figure  give  us  the 
numbers  that  we  have  already  found  to  be  the  powers  of  11. 
This  figure  is  called,  from  the  name  of  its  illustrious 
inventor,  the  arithmetical  triangle  of  Pascal,1 

1  Blaise  Pascal,  French  scholar  and  man  of  letters,  born  at  Clermont- 
Ferrand  (1623—1662). 


1 

1  1 

1   2 

1 

3 

3 

1 

4- 

6 

4 

1 

5 

10 

10 

5 

1 

6 

15 

20 

15 

6 

1 

7 

21 

35 

35 

21 

03 

1 

1 

1 

1 

1 

1    1 

2 

3 

4 

5 

6 

7    8 

3 

6 

10 

15 

21 

2836 

4- 

10 

20 

35 

56 

84 

5 

15 

35 

70 

6 

21 

56 

7 

28 

84 

8 

35 

FIG.  70. 


ARITHMETICAL  TRIANGLE  AND   SQUARE     81 

The  same  numbers  appear  in  a  figure  which  is  in  no 
way  different  to  the  preceding  one  except  by  its  arrange- 
ment (Fig.  70).     The  figure  1  is  written  in  each  of  the  divi- 
sions of  a  first  row  and  in  those  of 
the    first    column    of    a    piece    of 
squared  paper.     Afterwards  all  the 
other  divisions  are  filled  successively 
by  putting  in  each  the  sum  of  the 
number  which  can   be   read   above 
and  the  one  which  can  be  read  at 
the  left. 

Here  the  figures  1,  1  ;    1,   2,  1  ; 
1,  3,  3,  1  ;  ...  appear,   no  longer 
in  the  horizontal  lines,  but  in  those  which  are  oblique 
ascending  from  left  to  right. 

This  (Fig.  70)  is  called  the  arithmetical  square  ofFermat.1 

If  we  consider  (Fig.  71)  an  ordinary  chess-board,  the  left 

corner  division  O  and  any  division  whatever  X,  we  can 

ask  ourselves  by  how  many  differenf  ways  it  is  possible 

•  to   go   from   O  to  X  without  going 

back,  that  is  to  say,  moving  always 
from  left  to  right  and  downwards. 
The  arithmetical  square  of  Fermat 
gives  us  the  answer  if  we  lay  it  on 
the  chess-board.  Thus  for  Fig.  71 
as  it  is  drawn  we  have  84  different 
ways  of  reaching  X  from  O. 

The  numbers  of  these  figures  possess 


FIG.  71. 


many  very  curious    properties, 
mature  to  consider  them  now. 


But   it   would  be  pre- 


1  Pierre  Fermat,  French  mathematician,  born  at  Beaumont  de  Lomagne 
(1601 — 1665).  He  possessed  probably  the  most  powerful  genius  from 
an  arithmetical  point  of  view  that  has  ever  been  known. 


M. 


82  MATHEMATICS 


32.  Different  Ways  of  Counting. 

When  we  began  (Section  3)  to  make  numbers  by 
means  of  little  sticks,  then  bundles,  faggots,  etc.,  which 
leads  to  numeration,  we  could  equally  well  take  any 
other  number  than  10  little  sticks  to  make  a  bundle. 

For  example,  we  might  have  arranged  that  8  little 
sticks  would  make  a  bundle,  8  bundles  a  faggot,  and  so  on. 
The  result  would  have  been  that  the  figures  necessary 
to  write  any  number  (Section  10)  would  have  only  been 
1,  2,  3,  4,  5,  6,  7,  to  which,  of  course,  the  nought  would 
have  to  be  added. 

Such  a  method  of  writing  numbers  is  what  is  called 
a  system  of  numeration,  and  the  number  chosen  is  called 
the  base  of  this  system. 

Thus  the  system  of  which  we  have  so  far  made 
use,  which  is  universal,  is  called  the  decimal  system, 
and  has  for  its  base  10.  The  one  which  we  have  just 
indicated  would  have  8  for  its  base,  and  might  be  called 
the  octesimal  system. 

Supposing  that  12  were  taken  as  the  base  of  a  system, 
it  would  be  called  the  duodecimal  system,  and  it  would 
take  12  little  sticks  to  make  a  bundle,  12  bundles  to  make 
a  faggot,  and  so  on.  There  would  have  to  be  then, 
excluding  the  nought,  eleven  figures,  that  is  to  say,  the 

9  of  the  decimal  numeration,  and  two  others  to  represent 

10  and  11. 

When  a  numeration  system  has  for  its  base  a  number 
B,  it  always  requires  B — 1  figures,  without  counting  the 
nought,  and  the  number  B  is  invariably  written  as  10. 

It  is  useful  to  know  how  to  put  down  a  figure  in  one 
system  of  numeration  when  it  is  given  to  you  written  in 
another,  and  really  it  is  perfectly  easy. 

For  instance,  take  374,  written  in  the  system  with  a 
base  8.  We  will  try  to  write  it  in  the  decimal  system. 


DIFFERENT  WAYS   OF  COUNTING          83 

If  we  think  of  our  little  sticks  we  can  see  that  the  number 

in  question  contains 

4  little  sticks        . .          . .          . .          . .        4 

7  bundles  of  8  little  sticks         . .          . .     56 

3  faggots  of  8  x  8  little  sticks. .          . .   192 

252 

In  practice  we  would  arrive  at  the  same  result  more 
quickly  by  starting  from  the  left  and  saying  :  3  faggots 
of  8  bundles,  plus  7  bundles,  gives  31  bundles  ;  31  bundles 
of  8  little  sticks  makes  248,  which,  with  4  added,  makes 
252. 

If,  on  the  contrary,  the  number  598  is  written  in  the 
decimal  system,  and  we  wish  to  have  it  expressed  in  the 
system  with  8  for  base,  there  will  be  nothing  easier  than 
to  take  away  8  as  many  times  as  we  can,  and  the  remainder 
will  be  the  last  figure  to  the  right.  So  we  divide  598  by 
8,  and  take  the  remainder,  6 ;  this  operation  gives  us  the 
number  of  bundles  of  8,  which  is  74 ;  dividing  by  8  we 
have  the  number  of  faggots,  9,  and  for  remainder,  2 
bundles  ;  2  is  the  2nd  figure.  Dividing  9  by  8  we  see 
finally  that  we  have  for  remainder  1  bundle  (1  is  the  3rd 
figure)  and  that  we  have  1  box  (1  is  the  4th  figure). 

This  will  be  expressed  thus  : 
598  |  8 
38       74  |  8 
6         29      (  8 
1         1 

and  1126  is  the  required  number,  written  in  the  system 
of  base  8. 

If  it  was  necessary  to  write  this  number  with  a  12  base, 
we  should  have 

598  [  12 
118     49  |  12 
10       1       4 

and  the  result  would  be  41(10),  representing  by  (10)  the 
figure  10  of  the  duodecimal  system. 

G2 


84  MATHEMATICS 

We  have  just  seen  that  374  (system  8)  is  expressed  as 
252  (decimal  system).  In  the  system  base  12  it  would 
be  written  190. 

We  go  from  one  system  to  another  at  will,  using  the 
decimal  system  as  an  intermediary. 

With  use,  the  pupil  can  calculate  in  any  system  what- 
ever, only  the  essential  point  is  this,  never  to  let  him 
forget  that  carrying  is  no  longer  done  by  tens,  but  by 
groups  of  B,  if  B  is  the  base  ;  this  naturally  calls  for  some 
practice. 

We  give  below  the  number  1000  in  the  decimal  numera- 
tion, written  in  the  numeration  systems  with  various 
bases  3,  4,  5,  ...  up  to  12. 

B  =  3  . .  . .  1101001 

4  . .  . .  33220 

5  ..  ..  13000 

6  ..  ..  4344 

7  ..  ..  2626 

8  ..  ..  1750 

9  ..  ..  1331 

10    1000 

11  ..     ..    82(10) 

12  ..     ..   6(11)4 

It  is  worthy  of  notice  that,  using  the  numeration  system 
with  base  3,  and  employing  negative  figures,  the  numbers 
reduce  themselves  then  to  0,  +  1»  •  •  !•  This  fact 
is  rendered  more  interesting  when  we  learn  that  it  can  be 
put  to  practical  use  in  certain  questions  relative  to 
hydraulic  lifts.1 

M.  Marcel  Deprez  (membre  de  1'Institut),  to  whom  we 
owe  the  transport  of  energy  by  electricity,  has  been  good 
enough  to  tell  me  of  a  curious  way  of  weighing  by  means 
of  a  balance.  Let  us  suppose  that  we  place  weights  in 
both  scales.  Given  these  conditions,  the  problem  pro- 

1  This  application,  in  a  previous  French  edition,  was  placed  at  the 
end  of  the  book,  under  the  title  Note  on  a  question  of  weighing, 


DIFFERENT  WAYS  OF  COUNTING          85 

posed  is  to  determine  a  system  of  weights  (a  single  weight 
of  each  kind)  starting  from  1  gramme,  let  us  say,  in 
such  a  manner  that  it  will  be  possible  to  balance  bodies 
weighing  1,  2,  3  .  .  .  grammes  up  to  a  determined  limit. 

We  see  that  with  the  2  weights  1  gramme  and  3  grammes 
we  can  weigh  up  to  4  grammes,  since  2  =  3  —  1,  and 
4  =  3  +  1. 

By  taking  the  3  weights,  1,  3,  9  grammes,  we  can  weigh 
up  to  13  grammes. 

In  general,  if  we  have  taken  the  n  weights  1,  3  .  .  .  , 

Q»    1 

3""1   grammes,    we   can   weigh   up    to  -         -  grammes. 

•  — 

For  instance,  with  the  7  weights,  1,  3,  9,  27,  81,  243,  729 
grammes,  we  can  weigh  from  1  up  to  1093  grammes. 

This  question,  as  we  might  note,  leads  back  to  the 
writing  of  successive  numbers  in  the  system  of  base  3 
by  utilising  negative  figures.  Then,  instead  of  the  figures 
1,  2,  we  use  1,  T;  and  1  shows  that  the  corresponding 
weight  ought  to  be  placed  in  the  second  pan  of  the  balance. 
[1  is  another  way  of  writing  —  1.] 

For  instance,  59  is  written  in  this  system  11111,  for 
59  =  81  -  27  +  9  -  3  -  1.  To  weigh  59  grammes 
we  will  put  the  weights  81  and  9  in  a  scale,  and  27,  3,  1  in 
the  opposite  scale  ;  adding  to  this  last  a  body  weighing 
59  grammes,  the  balance  will  be  in  equilibrium. 

It  may  be  interesting  to  add  here  some  observations 
on  Roman  numeration.  It  is  now  no  longer  used  except 
for  the  purpose  of  marking  the  hours  on  the  dials  of 
watches  or  clocks.  The  child  will  be  able  also  to  decipher 
the  dates  on  old  inscriptions  if  he  understands  it,  but  that 
is  all,  so  that  the  actual  mathematical  interest  is  of  a  very 
moderate  kind.  It  is  quite  different  from  the  teaching 
point  of  view.  I  must  content  myself  with  only  a 
summary  of  the  observations  which  M.  Godard  (the 
then  director  of  the  school,  'Ecole  Monge)  brought  to 
my  notice  many  years  ago. 


86  MATHEMATICS 

If  sticks  are  laid  in  order  on  a  black  table,  for  instance, 
taking  care  to  space  them  equally,  and  we  suddenly  ask 
anyone,  without  any  warning,  how  many  sticks  there 
are  in  a  certain  group,  the  answer  will  be  given  immediately 
if  the  group  contains  two,  three,  or  four ;  beyond  that, 
for  five  and  upwards,  there  would  have  to  be  a  preliminary 
rapid  operation  of  the  mind,  a  mental  decomposition  of 
the  number,  so  that  the  answer  would  no  longer  be  the 
result  of  visual  impression.  This  is  a  well-assured  fact, 
which  is  verified  by  many  experiments. 

On  the  other  hand,  the  number  five  plays  an  important 
part  in  Roman  numeration. 

We  are  inclined  to  ask  if  it  has  not  originated  from  the 
physiological  fact  that  we  have  just  pointed  out,  and  its 
symbols  of  expression  from  the  anatomical  disposition 
of  the  human  hand. 

The  numbers  one,  two,  three,  four  would  be  represented 
by  one,  two,  three,  four  fingers  raised  : 

i,  ii,  in,  mi. 

We  can  make  a  tolerably  good  imitation  of  the  shape 
of  the  letter  V  by  means  of  the  whole  hand  held  up,  the 
thumb  stretched  away  from  the  four  fingers. 

Ten  can  be  shown  by  the  joining  of  two  hands,  one 
upwards,  V,  the  other  downwrards,  \,  which  gives  the 
letter  X. 

Only  the  first  principles  of  Roman  numeration  have 
been  mentioned,  and  we  refrain  from  any  comment  on 
the  other  symbols  L,  C,  M.  .  .  .  It  is,  however,  useful 
to  notice  that  the  consecutive  repetition  of  the  same  sign 
more  than  four  times  is  to  be  avoided. 

To  obtain  the  numbers  between  five  and  ten  the  neces- 
sary units  follow  the  sign  V  : 

VI,  VII,  VIII. 

Similarly,  for  numbers  above  ten,  we  have 
XI,  XII,  XIII. 

It  is  probable  that  as  a  later,  but  still  very  early 
improvement,  the  idea  was  carried  out  of  showing  sub« 


BINARY  NUMERATION  87 

traction  by  placing  the  unit  (or  other  symbol)  to  the  left 
of  a  fixed  sign,  while  transferred  to  the  right  the  same 
figure  would  signify  addition.  Thus  the  expressions 

IV,  IX,  XL,  .  .  . 

give  us  five  less  one,  or  four  ;  ten  less  one,  or  nine  ;  fifty 
less  ten,  or  forty ;  there  are  others  of  a  similar  kind.  It 
is  remarkable  to  notice  here,  in  however  embryonic  a 
form,  this  first  attempt  at  graphic  translation  of  the 
sign  by  the  sense. 

These  observations  seemed  to  me  sufficiently  curious 
to  deserve  mention.  They  seem  to  result  in  the  idea 
that  Roman  numeration  was  one  with  a  base  5,  but 
incomplete,  since  in  it  different  symbols  were  not  used 
to  indicate  the  first  four  numbers,  and,  above  all,  because 
that  central  pivot  of  all  rational  numeration — that  nothing 
which  is  everything  in  arithmetic — that  invaluable  resource, 
the  nought,  is  lacking. 

33.  Binary  Numeration. 

We  have  seen,  in  the  last  section,  that  if  B  is  the  base 
of  a  system  of  numeration,  this  system  requires  B  —  1 
figures,  without  the  nought.  If  2  is  the  base,  only  one 
figure  \vill  be  required,  the  figure  1. 

The  idea  of  this  numeration,  in  which  all  numbers  are 
written  by  means  of  only  two  characters,  1  and  0.  seems 
to  belong  to  Leibnitz,1  although  it  is  said  that  the  Chinese 
made  use  of  it  in  ancient  times. 

For  ordinary  use  in  calculations  this  system  would 
be  inconvenient  because  of  the  length  of  its  expressions. 
Thus  the  number  1,000  in  the  decimal  numeration 
would  be  represented  in  the  binary  system  by  the  figures 
1111101000,  ten  in  all.  But  in  certain  scientific  appli- 
cations the  employment  of  the  binary  system  is  at  once 
useful  and  interesting.  As  well  as  this,  we  find  in  it 
the  explanation  of  various  games,  such  as  "  the  ring 

1  Leibnitz,  German  philosopher  and  mathematician,  bom  at  Leipzig 
(1646—1716). 


88 


MATHEMATICS 


puzzle  "  and  Hanoi  To.vcr,  and  also  a  little  drawing-room 
game  which  depends  upon  the  use  of  binary  numeration, 
and  of  which  Edward  Lucas  gives  us  a  description  in 
his  "  Arithmetique  amusante  "  under  the  name  of  "  The 
Mysterious  Fan" 

To  make  this  clear,  suppose  that  we  have  written  the 
31  first  numbers  in  binary  enumeration : 


1   1 

2    10 

12   1100 

22  10110 

3    11 

13   1101 

23  10111 

4   100 

14   1110 

24  11000 

5   101 

15   1111 

25  11001 

6   110 

16  10000 

26  11010 

7   111 

17  10001 

27  11011 

8  1000 

18  10010 

28  11100 

9  1001 

19  10011 

29  11101 

10  1010 

20  10100 

30  11110 

11  1011 

21   10101 

31   11111 

Now,  on  a  piece  of  cardboard,  A,  let  us  write  (decimal 
system)  all  of  these  numbers  which  end  in  1  (binary 
system) : 

A  B  C  D  E 


1 

2 

4 

8 

16 

3 

3 

5 

9 

17 

5 

6 

6 

10 

18 

7 

7 

7 

11 

19 

9 

10 

12 

12 

20 

11 

11 

13 

13 

21 

13 

14 

14 

14 

22 

15 

15 

15 

15 

23 

17 

18 

20 

24 

24 

19 

19 

21 

25 

25 

21 

22 

22 

26 

26 

23 

23 

23 

27 

27 

25 

26 

28 

28 

28 

27 

27 

29 

29 

29 

29 

30 

30 

30 

30 

31 

31 

31 

31 

31 

ARITHMETICAL  PROGRESSIONS  89 

On  a  second  piece  of  cardboard,  B,  we  write  down  also 
the  numbers  whose  2nd  figure,  starting  from  the  right,  is 
a  1  in  binary  numeration ;  then  the  same  (cardboard 
slips  C,  D,  E)  for  the  3rd,  4th,  and  5th  figures. 

If  you  give  these  five  slips  to  anyone,  and  ask  him  to 
think  of  a  number  thereon,  and  to  hand  you  the  slips 
on  which  his  number  is  written,  but  only  those — that  will 
tell  you  the  numbers  which  enable  it  to  be  written  in 
binary  numeration.  It  is  quite  easy  to  verify  that, 
in  order  to  find  the  number  selected,  we  have  only  to 
add  the  first  numbers  written  on  each  slip.  Suppose 
we  take  25  as  the  number  chosen  ;  the  slips  you  will 
receive  will  be  A,  D,  E,  beginning  with  the  figures  1,  8,  16  ; 
1  +  8  +  16  =  25. 

The  game  may  be  played  with  the  number  63  instead 
of  31,  using  6  slips  instead  of  5,  and  with  127  also  (this 
latter  requiring  7  slips).  The  apparent  divination  may  be 
rendered  still  more  mysterious  by  the  use  of  proper 
names.  It  is  only  necessary  to  make  a  list,  giving  a 
number  to  each  name,  and  writing  the  names  on  the 
slips,  remembering  that  the  various  slips  begin  with 
1,  2,  4,  8,  16,  32,  64. 

Anyone  can  make  this  group  of  slips  of  cardboard, 
as  far  as  7  for  instance,  and  even  if  the  performer  does 
not  gain  a  reputation  as  a  sorcerer,  at  least  there  will  be 
the  solid  satisfaction  of  having  plenty  of  practice  in 
rapid  and  accurate  mental  addition.  Naturally,  without 
both  quickness  and  accuracy,  the  aforesaid  reputation 
can  never  be  maintained. 

34.  Arithmetical  Progressions. 
Take  a  series  of  numbers, 

4     7     10     13 

for  example,  such  that  the  difference  between  two  con- 
secutive ones  is  always  the  same  : 

7  -  4  =  10  -  7  =  13  -  10  =  3. 


90 


MATHEMATICS 


Such  a  series  is  called  a  progression  by  difference  or 
an  arithmetical  progression. 

The  difference,  being  constant,  3  in  this  example,  is 
called  the  common  difference  of  the  progression. 

The  numbers  4,  7,  10,  13  are  the  terms  of  the  progression. 
Here  we  have  only  written  four,  but  we  could  have  as 
many  as  we  liked. 

It  may  be  pointed  out  that  the  series  of  integral  numbers 
1,  2,  3  ...  forms  an  arithmetical  progression  with 
common  difference  1,  and  the  series  of  the  odd  numbers 
1,  3,  5  ...  forms  an  arithmetical  progression  with  the 
common  difference  2. 

We  will  try  and  represent  graphically  (Fig.  72)  the 
progression  4,  7,  10,  13,  the  example  mentioned  above. 
On  squared  paper,  counting  4  divisions  on  the  first  line 


• 

• 

• 

• 

0 

o 

o 

0 

0 

0 

o 

0 

0 

o 

0 

o 

o 

o 

o 

o 

0 

o 

o 

0 

0 

0 

o 

0 

0 

0 

o 

o 

o 

o 

0 

0 

0 

o 

FIG.  72. 

we  place  black  counters  in  each  ;  taking  7  divisions  on 
the  2nd  line,  10  on  the  3rd,  13  on  the  4th,  we  fill  each 
division  with  a  black  counter. 

Then,  by  adding  4  divisions  to  this  last  line,  and  com- 
pleting the  rectangle,  we  see  that  this  rectangle  contains 
the  terms  of  the  progression  twice  over,  as  shown  by  the 
black  and  the  white  counters. 

We  verify  in  this  way  that  the  sum  of  extreme  terms  is 
the  same  as  that  of  two  terms  equidistant  from  the 
extremes.  Finally,  the  sum  of  the  terms  of  the  progression 
will  be  half  the  number  of  the  divisions  of  the  rectangle, 

17  X  4 
that  is  to  say  — ~ — . 

£ 

As  a  rule,  if  a  and  b  are  the  two  extreme  terms  and  if 


GEOMETRIC  PROGRESSIONS  91 

n  is  the  number  of  the  terms,  this  sum  is  expressed  by 
(a  +  b)n 
~2       ' 
If  a  =  1,  and  if  the  common  difference  is  1,  then  b  =  n, 

n(n  -f  1) 
and  we  have  -  — » -. 

31 

If  fl  =  1,  and  if  the  common  difference  is  2,  then 
6  =  2n  —  I  and  we  have  n2.  We  thus  find  the  results 
already  met  with  above. 

Notice  the  analogy  which  exists  between  the  formula 

— 2 above,  and  that  of  the  area  of  a  trapezium. 

If  r  is  the  common  difference  of  an  arithmetic  progres- 
sion, this  progression  can  always  be  represented  by 

a       a  +  r       a  +  2r  .  .  .  a  +  (n  —  l)r. 

35.  Geometric  Progressions. 
If  a  series  of  numbers 

2         6         18         54         162 

for  instance,  is  such  that,  on  dividing  each  of  them  by  the 
preceding  one,  the  same  quotient  results,  these  numbers 
form  a  geometric  progression  or  progression  by  quotient. 
The  constant  quotient  is  the  common  ratio  of  the  pro- 
gression. It  may  be  said  that,  in  a  geometric  progression, 
the  relation  of  one  term  to  the  preceding  one  is  constant, 
and  is  called  the  common  ratio.  In  the  example  given 
above,  the  ratio  is  3,  the  first  term  is  2,  and  the  number 
of  terms  is  5. 

The  numbers  1,  10,  100,  1,000,  ...  in  the  decimal 
system,  form  a  geometric  progression  with  common 
ratio  10.  The  same  numbers  written  in  a  numeration 
system  with  the  base  B  form  a  geometric  progression 
whose  common  ratio  is  B. 

The  common  ratio  can  be  a  fraction,  as  well  as  a  whole 
number.  If  it  is  greater  than  1,  the  terms  go  on  increasing 


92  MATHEMATICS 

without  end  ;  the  progression  is  then  called  increasing. 
If  the  ratio  is  less  than  1,  the  progression  is  a  decreasing 
one,  and  its  terms  go  on  diminishing  more  and  more. 

It  is  interesting  to  be  able  to  find  the  sum  of  the  terms 
of  a  geometric  progression.  Let  us  go  back  to  the 
example  given  above 

2         6         18         54         162. 

If  we  multiply  any  term  by  the  common  ratio  3, 
we  have  the  following  term.  If  it  is  multiplied  by 
3  —  1  or  2,  then  we  shall  have  the  difference  of  two 
consecutive  terms  : 

2  (3  -  1)  =  6  -  2,     6  (3  -  1)  =  18  -  6, 

18  (3  -  1)  =  54    -  18,     54  (3  -  1)  =  162  -  54, 

162  (3  —  1)  =  486  -  162. 

Adding  all  these,  if  the  sum  is  s,  we  shall  have  then 
s  (3  —  1)  =  486  —  2,  since  all  the  other  numbers  cancel  ; 


so  that  s  =          ~  =  242. 

o  —  —  1 

In  general,  when 

abc     .........     k 

is  the  progression  with  common  ratio  q,  of  which  we  wish 
to  find  the  sum  s  ;  if  we  take  it  a  term  further  I  =  kq, 
we  shall  have 

a  (q  —  1)  =  b  —  a,  b  (q  —  1)  =  c  —  b,  .  .  . 
k(q  —  1)  =  I  -  k, 

and  s  (q  —  I)  =  I  —  a;  s  —      _  .. 

If    the   progression   is    a    decreasing    one,    we    have 

—  ,  which  comes  to  the  same  thing. 
1  ~~  f 
Geometric    progressions    play    an    important    part    in 

calculation  ;    they  have  numerous  applications. 

Even  if  the  common  ratio  is  not  much  greater  than  1, 
if  the  number  of  the  terms  becomes  rather  high,  the  pro- 
gressions lead  to  numbers  of  an  enormous  size  ;  these 


THE   GRAINS   OF   CORN  93 

results  amaze  us  to  begin  with  unless  we  are  forewarned. 
We  will  quote  several  instances  in  the  following  sections. 
It  is  useful  to  notice  if  a  is  the  first  term  of  a  geometric 
progression,  q  the  common  ratio,  and  n  the  number  of 
the  terms,  the  progression  may  be  written  thus 
a    aq    aq2     ...     aqn~l. 

36.  The  Grains  of  Corn  on  the  Chessboard. 

The  inventor  of  the  game  of  chess  is  not  exactly  known  ; 
but  there  exists  on  this  subject  an  old  Hindoo  legend 
which  deserves  to  be  remembered. 

Enchanted  by  the  new  diversion,  the  monarch,  accord- 
ing to  this  legend,  caused  the  inventor  to  be  brought 
before  him,  and  invited  him  to  fix  for  himself  the  reward 
which  he  desired. 

"  Let  your  Highness  simply  deign,"  responded  the  man 
"  to  order  your  servants  to  give  me  a  grain  of  corn,  to  be 
placed  in  the  first  division  of  my  chess-board  ;  2  on  the 
2nd,  4  on  the  3rd,  and  continue  thus,  always  doubling,  to 
the  64th  division." 

The  modesty  of  such  a  request  struck  the  monarch  with 
astonishment,  so  we  are  told,  and  he  gave  orders  that  the 
request  should  be  satisfied  without  delay.  But  he  was 
still  more  amazed  when  later  he  was  made  aware  of  the 
absolute  impossibility  of  fulfilling  his  commands.  In 
order  to  have  produced  the  necessary  quantity  of  corn 
the  product  of  eight  harvests  would  have  had  to  be 
gathered,  always  supposing  that  the  whole  of  the  soil 
of  his  kingdom  had  been  sown  with  seed. 

The  number  of  the  grains  of  corn  required  is  the  sum 
of  the  terms  of  the  progression 

1     2     23  .  .  .  263, 

which  gives  2s4  —  1.     Here  is  the  number  written  in 
the  decimal  system : 

18446744073709551615. 
There  are  twenty  figures  in  this,  as  we  see.    We  will 


94  MATHEMATICS 

not  attempt  to  read  it.  The  words  which  we  would  utter 
would  not  convey  much  meaning  to  our  mind.  However 
we  shall  presently  find  others  much  larger. 

37.  A  very  Cheap  House. 

One  of  our  friends,  probably  knowing  the  chess-board 
story,  had  a  little  two-storied  house  built  for  him.  A 
flight  of  7  steps  led  from  basement  to  first  floor,  and  the 
staircase  which  led  to  the  second  floor  had  19  steps. 

At  the  end  of  several  years  he  decided  it  was  time  to 
put  his  little  place  on  the  market,  as  it  was  in  good 
order,  and  had  a  veiy  pleasant  aspect.  To  the  first 
would-be  purchaser  Smith  made  the  following  pro- 
position : 

"  I  am  not  at  all  unreasonable,  and,  moreover,  I  really 
wish  to  sell ;  suppose  I  offer  you  the  house  if  you  will  put 
a  cent  on  the  first  of  the  small  flight  of  steps,  2  on  the 
2nd,  4  on  the  3rd,  doubling  thus  on  every  step  till  the  end 
of  the  staircase  is  reached.  It  is  really  nothing,  there  are 
only  26  steps  in  all." 

"  Done  !  there's  my  hand  upon  it,"  cried  Jones,  the 
would-be  possessor,  beside  himself  with  joy  at  such  a  stroke 
of  good  luck. 

And  the  next  day  Jones,  having  first  entertained  Smith 
to  a  sumptuous  meal,  set  out  for  the  house  to  count  out 
the  required  coins  on  the  26  steps,  i.e.  226 — 1  cents. 

Up  to  the  top  of  the  first  flight  of  steps  all  went  well, 
and  the  first  few  steps  of  the  staircase  presented  no 
insuperable  difficulty  ;  but  soon  his  purse  emptied  much 
more  quickly  than  he  had  imagined  it  possible. 

The  vendor  very  obligingly  offered  to  tell  Jones  the  total 
amount  of  his  debt,  so  that  he  would  have  no  need  to 
go  up  further.  "  My  dear  Jones,"  he  cried,  "  you  owe 
me  $671,088.63,  but  from  an  old  friend  like  you,  I  will 
not  expect  the  .63,  I  shall  be  pleased  to  meet  you  that 
far." 

Poor    Jones'    face    lengthened    remarkably    at    this 


THE  INVESTMENT  OF   A    CENTIME         95 

announcement,  and  ever  since  he  has  insisted  that  each 
of  his  children  should  be  taught  what  a  progression  means. 
He  himself  is  perfectly  acquainted   with   its   nature, 
but  his  knowledge  was  rather  expensive. 

38.  The  Investment  of  a  Centime. 

One  of  the  most  important  practical  applications  of 
progressions  is  that  which  concerns  compound  interest. 
If  we  put  out  100  dollars  at  interest  for  a  year,  at  5  per 
cent.,  it  brings  in  5  dollars.  If,  instead  of  touching 
the  5  dollars,  we  join  them  to  the  100,  that  makes  105 
which  we  can  put  out  during  a  second  year,  and  so  on. 
When  the  number  of  years  becomes  considerable,  the 
growth  of  capital  by  this  operation  of  compound  interest 
is  absolutely  startling. 

Suppose,  for  example,  that  at  the  beginning  of  the 
Christian  era  a  cent  had  been  put  at  compound  interest 
at  the  rate  of  5  per  cent. ;  it  is  calculated  that  toward 
the  end  of  the  19th  century  its  acquired  value  would 
be  more  than  200  millions  of  spheres  of  pure  gold  as  big 
as  our  earth. 

We  may  say,  in  passing,  that  such  a  result  shows 
us  the  impossibility  of  an  absolute  application  of  com- 
pound interest  in  practice.  The  enormousness  of  the 
amount  forbids  any  exact  idea  of  such  a  sum. 

It  will  be  far  better  to  put  a  question  of  this  kind  : 
for  what  length  of  time  must  a  dollar  be  put  out  at  com- 
pound interest  at  the  rate  of  5  per  cent,  so  that  the 
acquired  value  may  be  a  hundred  million  dollars? 

The  answer  is  378  years,  so  that  if  one  of  our  ancestors, 
about  1527,  in  the  time  of  Henry  VIII.,  had  conceived  the 
brilliant  idea  of  placing  1  dollar  to  your  credit,  at  5  per 
cent,  compound  interest,  this  would  have  grown  to-day 
to  the  enormous  value  of  100  million  dollars. 

If  the  same  thing  had  been  done  in  the  year  59  of  our 
era,  at  the  rate  of  only  1  per  cent.,  the  same  result  would 


96  MATHEMATICS 

have  been  obtained  in  1907,  that  is  to  say,  the  dollar 
thus  placed  would  to-day  be  worth  100  million  dollars 
(always  supposing  no  accidents  occurred  in  the  interval!). 

39.  The  Ceremonious  Dinner. 

One  evening  twelve  people  had  arranged  to  dine 
together.  Each  of  them  attached  great  importance  to 
points  of  etiquette  ;  now  the  seating  of  the  party  had  not 
been  arranged  in  advance,  and  a  courteous  discussion 
arose  at  the  moment  of  going  to  table  which,  however, 
did  not  lead  to  any  result.  Some  one,  as  a  means  of 
solving  the  difficulty,  proposed  that  all  the  possible  ways 
of  attacking  the  problem  should  be  tried  ;  there  would 
be  nothing  to  do  then  but  choose  the  one  which  seemed 
the  best.  Accordingly  this  was  done  for  some  few 
minutes,  but  they  became  so  mixed  up  that  it  did  not 
seem  to  hold  out  any  satisfactory  prospect.  Happily, 
among  the  guests,  there  was  one,  a  professor  at  the  college 
in  the  town,  who  was  a  mathematician.  "  My  good 
friends,"  said  he,  "  the  soup  is  going  cold.  Let  us  seat 
ourselves  at  random ;  that  will  be  quickest."  This  wise 
counsel  was  followed  and  the  repast  was  brought  to  a 
close  amid  the  greatest  cordiality.  At  dessert,  taking  up 
the  subject  once  more,  "  Do  you  know,"  said  the 
professor,  "  how  long  it  would  have  taken  us  to  try  all 
the  possible  ways  of  seating  ourselves  round  this  table, 
taking  no  more  than  just  one  second  to  move  from  one 
seat  to  another  ?  "  and,  as  each  kept  silence,  he  went 
on  to  say,  "  Continuing  this  little  game,  day  and  night, 
without  stopping  a  single  moment,  we  should  have 
been  15  years  and  2  months,  taking  no  notice  of  leap 
years.  You  see  that  the  meat  would  have  dried  up  and 
we  ourselves  should  all  have  died  of  hunger,  weariness, 
and  loss  of  sleep.  By  all  means  let  us  be  ceremonious 
if  we  wish,  but  not  to  excess." 

This  was  absolutely  correct ;    the  precise  number  of 


THE   CEREMONIOUS  DINNER  97 

different  ways  in  which  12  people  can  take  their  places 
at  a  table  laid  with  12  covers  is  just  479,001,600  ;  more 
than  479  millions,  as  you  see. 

This  result  is  amazing  when  we  reflect  that  for  2  diners 
2  seconds  of  time  only  would  have  been  needed ;  and 
even  for  4  the  trials  might  have  been  made  in  less  than 
half-a-minute. 

The  enormous  numbers  we  have  just  mentioned  are 
due  to  permutations,  and  the  deduction  is  easy  to  make. 

When  several  different  objects  are  in  question,  which 
can  be  arranged  in  various  ways  indicated  beforehand, 
any  particular  arrangement  adopted  is  a  permutation  of 
these  objects. 

If  we  deal  with  two  objects  a,  b  and  with  two  different 
places,  the  only  two  permutations  possible  are  a  b  and  b  a. 

To  form  the  permutations  of  three  objects,  a,  b,  c,  we 
can  take  the  permutation  a  b  and  join  c  to  it  at  three 
different  places ;  it  can  come  after  b,  between  a  and  b, 
or  before  a. 

The  permutation  b  a  will  also  give  three  by  joining  c 
to  it ;  so  that  we  shall  have  the  table  of  permutations 
of  a,  b,  c  by  writing 

ab c  b  a  c 

a  c  b  b  c  a 

cab  c  b  a 

and  this  gives  2x3  =  6  permutations. 

If  we  take  any  one  of  these  permutations,  a  b  c  for 
instance,  and  add  on  to  it  a  4th  letter,  we  shall  have 
then  4  permutations 

abed         abdc         adbc         dabc 

and  each  permutation  of  3  letters  thus  furnishing  4  of 
4  letters,  the  number  of  permutations  of  4  letters  will 
be  6  x  4,  or  2  .  3  .  4  =  24. 

Continuing  thus,  the  number  of  permutations  of  5 
letters  would  be  2.3.4.5;  and  generally,  that  of  the 

3f  H 


98  MATHEMATICS 

permutations  of  n  letters  will  be  2  .  3  .  4  .  .  .  n.     This 
is  often  represented  by  n  /  or  j^.1 

We  can  see  below  how  rapidly  these  n  !  numbers  of 

permutations  grow  when  n  increases. 

n  n! 

2  2 

3  6 

4  24 

5  120 

6  720 

7  5040 

8  40320 

9  362880 

10  3628800 

11  39916800 

12  479001600 

Permutations  play  a  most  important  part  in  mathe- 
matics.    They  can  be    used,    besides,  in  various  games 
and  amusements,  such  as  anagrams.      Very  many  papers, 
exceedingly  learned  ones,   have  been  published  on  per- 
mutations.    We  shall  not  deal  with  them  here,  but  may 
mention  the  happy  idea  of  Ed.  Lucas,  of  representing  by 
a  drawing  the  permutations  of  several  objects.     He  has 
called  it  pictured  permutations.     To  give  the  pupil  a  clear 
understanding  of  this  idea,   suppose  that  we    make  on 
squared  paper  a  square  of  n  columns,  of  n  rows  each  ; 
and,  confining  ourselves  to  permutations  of  4  objects, 
n  shall  equal  4.     There  will  be  a  square  of  16  divisions. 
If  we  replace  the  4  objects  a,  b,  c,  d  by  the  4  numbers 
1,  2,  3,  4,  the  permutation,  c  b  d  a,  for  example,  can  be 
written  3241,  and  so  with  the  others.     Taking,  then,  the 
first  column  of  the  square,  we  mark  the  3rd  division, 
and  shade  it,  the  same  for  the  2nd  division  of  the  2nd 
column,  the  4th  of  the  3rd  column,  and  the  1st  of  the 
4th.      The   four  divisions  so  shaded  thus  stand  for  the 
permutation  c  b  d  a. 

Fig.  73  shows  us  the  24  permutations  of  4  objects. 

«  /  is  called  "  factorial  n  "  because  it  is  made  up  of  factors. 


THE  CEREMONIOUS  DINNER 


99 


To  make  this  easy  to  understand  we  give  below  the  table 
of  permutations  which  corresponds  to  the  figure,  in  the 
same  order. 


abed 
acbd 
cabd 
bacd 
bead 
chad 


FIG.  73. 

abdc  adbc 

acdb  adcb 

cadb  cdab 

bade  bdac 

bcda  bdca 

cbda  cdba 


dabc 
dacb 
dcab 
dbac 
dbca 
dcba 


H.2 


100  MATHEMATICS 

If  we  consider  any  one  of  the  squares  of  Fig.  73  as  a 
chess-board,  the  shaded  divisions  represent  the  positions 
of  castles  which  cannot  be  taken  by  one  another,  and 
that  applies  to  all  analogous  squares.  It  follows,  then, 
that  on  an  ordinary  chess-board  of  64  divisions  we  can 
place,  in  40,320  (8 !)  different  ways,  eight  castles  so  that 
they  cannot  take  one  another.  On  a  board  of  100  divisions, 
ten  castles  could  be  placed,  under  the  same  conditions, 
in  3,628,800  (10 !)  different  ways.  We  can  consult  the 
table  given  on  page  98  for  these  numbers.  Questions  of 
this  kind  are  not  easy  to  answer  without  the  help  of 
permutations.  With  their  aid  the  solutions  are  perfectly 
simple. 

You  can  also  ask  yourself  in  how  many  different  ways 
we  can  place  the  cards  in  a  game  of  piquet ;  the  answer  is 
32  !,  or  those  in  a  game  of  whist,  which  will  be  52 !,  but  I 
do  not  advise  you  to  try  and  write  these  numbers  in  the 
decimal  system.  Try  rather  to  find  the  time  it  will  take 
to  carry  out  all  these  changes,  taking  a  second  to  do  each 
one.  This  is  a  pleasure  I  am  leaving  to  my  readers, 
or  rather  to  their  pupils.  But  let  them  not  try  to  write 
these  numbers,  even  counting  them  in  centuries,  for  such 
an  effort  would  hardly  tend  to  the  education  of  one's 
mind. 

40.  A  Huge  Number. 

We  are  raising  ourselves,  by  progressions  on  the  one 
hand,  and  permutations  on  the  other,  to  great  heights 
on  the  ladder  of  numbers. 

In  the  hope  of  returning  to  more  reasonable  limits, 
and  thus  escaping  a  feeling  of  giddiness,  ask  some  one 
to  write  down  the  largest  number  possible  by  using  three 
9's.  Generally  the  answer  will  be 

999, 

quite  a  modest  number,  indeed,   which  does   not  make 
one's  brain  whirl  in  the  slightest. 


A  HUGE  NUMBER  101 

But  if  by  chance  you  have  happened  upon  a  conscien- 
tious mathematician,  anxious  to  give  you  an  absolutely 
correct  answer,  you  will  read,  by  a  slight  change  in  the 
position  of  the  9's, 


This  means  that  we  must  raise  9  to  a  power  marked  by 

the  number  99.  This  last  is  easily  found  in  a  few  minutes. 
Your  pupil  will  certainly  give  it  to  you  without  any 
hesitation  if  you  do  not  care  about  working  it  yourself. 
It  is 

387,420,489, 

and  this  result  is  very  interesting,  for  you  know,  thanks 
to  the  pupil's  answer,  that  all  you  have  to  do  to  obtain 
the  required  number  in  the  decimal  system  is  to  make 
387,420,488  multiplications. 

These  are  very  simple,  having  only  9  as  multiplicator, 
but  their  number  rather  inspires  hesitation. 

Decidedly,  I  cannot  encourage  you  to  undertake  the 
task.  I  will  only  tell  you  —  so  that  it  can  be  repeated  to 
the  pupil,  who  can  verify  it  later  —  that  the  number 

99 
9   ,  if  written  in  decimal  numeration  would  have 

369,693,100  figures. 

To  write  it  on  a  single  strip  of  paper,  supposing  that 
each  figure  occupied  a  space  of  i-inch,  the  length  of  the 
strip  would  need  to  be 

1,166  miles,  1,690  yards,  1  foot,  8  inches, 
which  is  farther  than  from  New  York  to  Chicago. 

Under  the  same  conditions,  to  write  1010  ,  we  would 
need  a  strip  of  paper  long  enough  to  encircle  the  earth.1 

1  This  remark  was  made  by  M.  Ch.  Ed.  Guillaume,  in  a  very 
interesting  article  in  the  Revue  Gtn&rale  dea  Sciences  (,30th  Oct.,  1906). 


102  MATHEMATICS 

The  time  that  we  should  spend  in  writing  down  the 

9 

number  99  ,  taking  a  second  for  each  figure,  and  working 
ten  hours  per  day,  would  be  approximately  28  years 
and  48  days,  working  continuously  without  stopping  for 
Sundays  and  holidays. 

To  add  to  your  information,  I  can  assure  you  that  the 
first  figure  of  the  number  we  are  seeking  is  a  4,  and  that 
the  last  is  a  9.  That  leaves  us  just  369,693,098  figures  to 
find.  Perhaps  you  may  think  this  but  a  paltry  assistance, 
and  I  am  of  the  same  opinion.  However,  I  hope  you  will 
agree  with  me  that  the  title  I  have  chosen  for  this  section, 
"  A  Huge  Number,"  is  thoroughly  justified. 

I1  22 

It  is  worth  noticing  that  1     is  simply  1,  that  2     =16, 

a« 
and  that  3'    is  a  number  of  13  figures, 

7,625,597,484,98r.1 


$1.  The  Compass  and  Protractor. 

In  the  various  drawing  exercises,  which  the  pupil 
ought  never  to  have  been  allowed  to  discontinue,  circles 
or  fragments  of  such  may  have  occurred  which  were 
drawn  almost  freehand. 

1  In  spite  of  the  explanations  given,  several  readers  have  confused 

3» 
the  signification  of  3   ,  and  several  have  written  to  me  saying  that  they 

made  the  result  19683,  and  not  a  number  of  13  figures.  This  arises 
from  a  false  interpretation  of  the  symbol  a ' ,  which  can  be  read 
(  a  )  or  a  (  h  ).  This  last  interpretation  is  the  only  reasonable  one  ; 

/    b\e  be 

for  (  «    )  =  a    .      It   would   be   illogical    to   employ   the    expression 

,c  be  9  /  9\ 

a   to  represent  the  more  simple  one  a    .    So  99  can  only  signify  9l  9 

3 

and  33  means  327  and  not  273. 


THE   COMPASS  AND  PROTRACTOR      103 

For  drawings  in  which  we  need  a  certain  amount  of 
precision,  the  time  has  come  to  accustom  the  pupil  to  the 
use  of  the  compass.  He  must  practise  by  tracing  arcs 
of  circles  first,  then  whole  circles,  in  pencil  to  begin  with, 
and  in  ink  afterwards.  He  will  be  shown,  following  the 
mode  pointed  out  in  all  the  classical  treatises,  how  to  draw 
perpendiculars  to  straight  lines,  to  construct  angles,  and 
various  other  figures,  etc. 

These  constructions,  when  they  are  required  to  be  exact, 
ought  besides  to  admit  of  the  use  of  the  protractor,  which 
is  as  simple  in  its  use  as  the  compass,  and  is  of  rather 
similar  service  in  the  diverse  forms  it  assumes ;  semi- 
circular or  rectangular,  made  of  metal,  horn,  etc. 

As  to  the  method  of  division  of  the  protractor,  prefer- 
ence must  be  given  to  that  in  grades,  where  the  right  angle 
is  divided  into  100  grades,  and  the  grade  then  into  tenths 
and  hundredths,  etc. 

This  method  of  measuring  angles  was  instituted  at  the 
same  time  as  the  metric  system.  Then  it  was  abandoned 
for  the  old  system  of  degrees,  minutes,  etc. 

In  France  they  are  now,  and  with  good  reason, 
returning  to  the  grade  method,  even  in  various  official 
lists  ;  and  several  important  public  offices  constantly 
make  use  of  this  division  into  grades.  It  is  most 
advisable,  therefore,  to  make  it,  from  the  outset,  quite 
familiar  to  the  pupils,  and  to  show  them  the  half  of 
a  right  angle  under  the  name  of  50  grades  (rather  than 
45  degrees). 

These  constructions  are  to  be  made,  and  should  be  very 
simple.  Indeed,  they  may  often  be  left  to  the  initiative 
of  the  pupil,  remembering,  however,  that  it  is  important 
to  make  him  execute  the  same  construction  by  means 
of  different  scales.  He  will  conceive  thus  the  notion 
of  figures  having  the  same  shape  but  different  size, 
that  is  to  say,  similar  figures,  without  being  taught  any 
definition. 

The  child   will  perceive  very  quickly  that  the  scale 


104  MATHEMATICS 

chosen  to  make  a  construction  will  cause  no  change 
in  the  angles  ;  but  on  the  contrary,  if  a  double  or  treble 
scale  be  adopted,  all  the  corresponding  lengths  will  be- 
come doubled  or  trebled.  In  short,  without  making  any 
geometrical  study  as  far  as  the  present  is  concerned,  he 
will  acquire  a  knowledge,  born  of  experience,  of  many 
truths,  whose  proof  will  be  so  much  the  more  easily 
assimilated  later. 

There  are  certain  other  properties  useful  to  know,  and 
certain  names  useful  to  retain  hi  the  memory,  for  which 
the  pupil  must  provisionally  take  your  word,  and  give  you 
credit.  These  form  the  subject  of  the  following  sections. 

32.  The  Circle. 

The  circle  is  the  round  figure  (Fig.  74)  that  is  traced 
with  a  compass,  one  of  the  points  remaining  fixed.  The 
point  O  which  is  fixed  is  the  centre  ;  the 
distance  from  the  centre  to  any  point  M 
of  the  circle  is  called  the  radius.  Twice 
the  radius  is  the  diameter;  any  straight 
line,  MM',  passing  through  the  centre, 
is  a  diameter;  the  length  of  the  segment 
MM  is  double  that  of  the  radius.  When 
we  take  the  two  points  A,  B  on  a  circle,  that  portion 
of  the  circle  limited  to  A  and  B,  whether  on  one  side  or 
the  other,  is  called  an  arc  of  the  circle.  The  straight  line 
AB  is  a  chord  which  subtends  the  two  arcs  AB.  The 
space  between  the  chord  and  the  arc  is  called  a  segment 
of  a  circle.  When  the  point  O  is  joined  to  two  points 
A,  B  of  the  circle,  the  angle  AOB  is  called  an  angle  at 
the  centre.  The  angle  AMB,  whose  sides  pass  through 
AB,  and  whose  corner  is  at  M  on  the  circle,  is  an  angle 
inscribed  in  the  segment  AMB  ;  this  angle  is  the  half  of 
the  angle  AOB.  When  we  join  the  centre  to  the  middle 
C  of  the  arc  AB,  the  straight  line  OC  is  perpendicular 
to  the  chord  AB,  which  is  cut  in  half  at  D. 


THE  CIRCLE  105 

If  we  take  a  point  N  on  the  circle,  below  the  chord  AB 
on  the  figure,  the  sum  of  the  two  angles  AMB,  ANB  is 
equal  to  two  right  angles. 

When  we  consider  (Fig.  75)  a 
circle  and  a  straight  line,  this  last 
may  be  (DJ  outside  the  circle,  or 
(D2)  may  cut  it  at  two  points  ;  this 
is  said  to  be  a  secant ;  or,  finally, 
(D8)  may  touch  the  circle  at  one 
point  only,  in  which  case  it  is  a 
tangent  to  the  circle.  The  distance  FIG.  75. 

from  the  centre  to  the  straight  line  is 

greater  than  the  radius  for  an  exterior  straight  line, 
less  „       „        „        „    a  secant. 

equal  to       „        „        „   a  tangent. 
The  point  common  to  the  tangent  and  to  the  circle 
is  the  point  of  contact.     The  tangent  is  perpendicular  to 
the  radius  which  is  drawn  to  the  point  of  contact. 

In  any  circle  whatever  there  is  the  idea  of  length  ;  this 
would  be  that  of  an  extremely  fine  thread  which  would 
surround   the  entire   ring.     Although  this  general   idea 
lacks  precision,  it  presents  a  picture, 
and  conveys  an  impression  to  the 
mind;    it  will  take  definite  shape 
later.     The  length  of  the  circle  of 
which    we    have    just     spoken   is 
called  its  circumference. 

If  (Fig.  76)  we  consider  any  two  circles,  O,  O',  the 
ratio  of  the  circumferences  is  equal  to  the  ratio  of  the 
diameters.  This  means  that  the  ratio  of  the  length  of 
the  circumference  to  that  of  the  diameter  is  the  same 
in  each  of  the  two  circles,  and  therefore  in  all  circles. 
This  ratio  of  the  circumference  to  the  diameter,  which 
cannot  be  exactly  expressed  by  any  fractions  whatever, 
is  greater  than  3-14,  but  less  than  3-1416;  it  is  indi- 
cated by  the  Greek  letter  IT.  For  many  ordinary  purposes 
3-14  is  sufficiently  near,  and  3-1416  will  be  found  exact 


106  MATHEMATICS 

enough  in  almost  every  case  when  greater  precision  is 
needed. 

If  C  is  the  length  of  the  circumference,  and  if  D  =  2R 

C 
is  the  diameter,  R  being  the  radius,  the  ratio  ^  is  then 

TT.  This  means  that  C  =  irD  =  27rR.  In  ordinary 
practice,  C  =  3'14  x  D  =  6-28  x  R. 

This  tells  us  easily  the  circumference  of  any  circular 
object,  when  we  know  the  radius  or  the  diameter,  and 
also  how  to  find  the  diameter,  for  instance,  of  a  round 
tower,  of  the  trunk  of  a  tree,  or  of  a  column,  when  the 
circumference  can  be  measured  with  a  narrow  tape  or 
in  any  other  way. 

Direct  the  children's  attention  as  far  as  possible  to  the 
advisability  of  doing  these  exercises  on  real  objects  ;  do 
not  neglect  the  opportunity  of  making  them  check  the 
approximate  value  of  TT  which  they  have  used,  when,  at 
one  and  the  same  time,  the  circumference  and  the  diameter 
can  be  measured. 

43.  The  Area  of  the  Circle. 

Just  as  we  acquire,  by  intuition,  a  knowledge  of  the 
circumference  of  the  circle,  so  also  we  feel  that  the  portion 
of  space  inside  the  line  has  a  certain  breadth,  an  area 
which  we  should  be  able  to  measure.  It  is  found  that 
this  area  can  be  obtained  by  multiplying  the  length  of 
the  circumference  by  half  the  radius.  And,  as  we  have 
seen  that  C  =  27rR,  it  follows  that  the  area  S  =  irR2, 


or  again  that  8  =  7  D2.  ^  happens  thus  that  the  areas 

~fc 

of  two  circles  have  between  them  a  relation  which  is  the 
same  as  that  of  the  squares  of  the  radii  or  of  the  diameters. 
Here  again,  practical  examples,  as  varied  as  possible, 
will  serve  as  subjects  for  exercises  on  these  questions  of 
areas:  circular  masses  of  stone  in  a  garden,  fountains 
in  parks,  the  floor  of  a  riding  school  which  is  to  be  covered 


CRESCENTS  AND   ROSES 


107 


B 


FIG.  77. 


with  sand,  the  picture  of  a  round  table  ;   measuring  rings 
or  halos  by  the  difference  between  two  circles,  etc.,  etc. 

H.  Crescents  and  Roses. 

In  the  greater  number  of  the  treatises  on  drawing, 
models  of  figures  made  up  of  circular  elementary  forms 
are  frequently  formed  which  can  be  traced  with  the  help 
of  the  compass  and  make  interesting  exercises. 

Merely  as  specimens,  we  shall 
show  here  a  small  number  only  of 
these  figures,  of  which  some  are 
very  well  known. 

If  we  draw  (Fig.  77)  a  semi- 
circle having  for  diameter  BC,  and 
if  we  take  a  point  A  anywhere  on 
this  line,  the  triangle  ABC  is  always 
right-angled,  the  angle  A  being  a 
right  angle.  Now,  let  us  describe  two  other  semicircles 
on  AB,  and  on  AC  as  diameters.  We  shall  thus  have 

two  kinds  of  crescents  (shaded 
on  the  figure).  What  makes 
this  figure  interesting  is  that 
the  sum  of  the  areas  of  the  two 
crescents  is  exactly  equal  to 
the  area  of  the  triangle  ABC. 
This  property  was  known  at 
the  time  of  the  Grecian  era,  and 
the  construction  that  we  have 
just  indicated  has  become 
classical  under  the  name  of 
the  crescents  of  Hippocrates.1 

Another  interesting  construction  is  that  shown  in  Fig.  78. 
Let  us  divide  the  diameter  AB  of  a  circle  into  five  equal 
parts  by  the  points  C,  D,  E,  F.  On  AC,  AD,  AE,  AF 
as  diameters  we  draw  semicircles  above  the  line ;  then 

>  Hippocrates  of  Ohio,  Greek  geometrician,  5th  century  B.C. 


FIG.  78. 


108 


H 


on  CB,  DB,  EB,  FB  we  draw  semicircles  below.  By 
means  of  these  circular  lines,  the  circle  is  divided  into  five 
parts  which  have  the  same  area.  Instead 
of  five,  any  other  number  n  could  be 
taken.  It  would  be  sufficient  to  divide 
the  diameter  AB  into  n  equal  parts. 

In  a  circle  (Fig.  79)  let  us  take  two 
diameters  perpendicular  to  one  another 
AB,  CD.  Having  formed  the  square 
OBEC,  let  us  trace  from  B  to  C  a 
quarter  of  the  circle  of  which  E  is  the 
centre,  tracing  at  the  same  time  three 
other  quarter  circles  CA,  AD,  BD ; 
and  the  whole  (the  shaded  part)  forms  a  sort  of  star  with 
four  points.  The  area  of  this  star  is  (4  —  ?r)  R2,  or  nearly 
0-86  x  R2. 

If  (Fig.  80)  we  again  take  two 
diameters  perpendicular  to  one  another, 
AB,  CD,  and  if  we  draw  the  semicircles 
having  for  diameters  BC,  CA,  AD,  DB,  A| 
we  obtain  a  rose  with  four  leaves.  The 
area  of  this  rose  is  (TT  —  2)  R2,  or  nearly 
1'14  x  R2,  the  radius  being  always  repre- 
sented by  R. 

Opening  the  compass  to  a  length  equal 
to  the  radius,  and  marking  out  this  length  (Fig.  81) 
successively  on  the  circle  at  B,  C,  .  .  .  we  shall  find  that 
it  falls  once  more  on  the^  point  A  after 
the  6th  operation.  If,  with  B,  D,  F  as 
centres,  with  the  radius  R,  we  describe 
the  arcs  of  the  circles  AC,  CE,  EA,  which 
all  pass  through  the  centre,  a  rose  with 
three  leaves  will  be  produced. 

By  tracing  (Fig.  82)  with  the  same 
construction  the  six  arcs  of  the  circle 
with  the  centres  A,  B,  C,  D,  E,  F  we  obtain  a  rose  with 
six  leaves. 


FIG.  81. 


CRESCENTS  AND  ROSES 


109 


We  will  limit  ourselves  simply  to  these  few  illustrations, 
given  only  by  way  of  example.  In 
practice,  they  should  be  constantly  varied, 
and  we  should  impel  the  child  to  use 
his  own  imagination  to  form  fresh  figures. 
This  will  follow  as  a  matter  of  course,  for 
as  soon  as  ever  he  becomes  at  all  familiar 
with  the  use  of  the  compass  and  other 
elementary  drawing  instruments,  he  will 
take  a  pride  in  forming  various  figures, 
and  will  bestow  his  time  and  attention  on  it. 


FIG.  82. 


45.  Some  Volumes. 

If  it  is  important  to  determine  the  areas  of  surfaces, 
it  is  just  as  necessary,  in  practice,  to  ascertain  the  volumes 
of  bodies.  To  do  this,  we  must  have  a  unit  of  volume, 
just  as  for  determining  lengths  we  require  a  unit  of 
length,  and  for  areas  a  unit  of  area.  This  unit  of 
volume  is  always  the  volume  of  a  cube  having  for  its  side 
the  unit  of  length. 

Starting  from  that  point,  the  volumes 
of  a  certain  number  of  bodies  of  regular 
shapes  are  found  by  very  simple  means. 

We  intend  summarising  these  now,  in 
such  a  manner  that  there  will  then  be  no 
difficulty  in  solving  certain  ordinary 
Questions.  First  of  all,  let  us  recall  to  our 
minds  the  bodies  that  we  have  already 
defined,  and  also  point  out  three  others 
which  are  frequently  to  be  encountered 
in  ordinary  practice. 

We  have  seen  what  a  cube  is,  also  a  parallelepiped, 
a  prism,  and  a  pyramid. 

In  all  these  bodies,  we  only  see  straight  lines  and  planes  ; 
generally  we  call  them  polyhedra.  In  the  three  others  of 
which  we  are  now  going  to  speak  this  is  no  longer  the  case. 


FIG.  83. 


110  MATHEMATICS 

Let  us  imagine  (Fig.  83)  that  a  rectangle  AGO' A' 
turns  round  its  side  OO'  ;  it  thus  makes  a  body  which  is 
called  a  rectangular  cylinder.  A  hat  or  muff  box,  a 
lamp-glass,  the  inside  of  a  pint  pot  (sometimes),  will 
show  the  general  form  of  a  cylinder.  The  two  sides  OA, 
O'A'  describe  two  circles  of  equal  radius  OA  =  O'A', 
which  are  called  the  bases  of  the  cylinder;  OO',  which  has 
not  moved,  represents  the  distance  apart  of  the  planes 
of  the  two  bases  :  this  is  the  height  of  the  cylinder. 

Let  there  be  (Fig.  84)  a  rectangular  triangle  AOS,  in 
which  O  is  a  right  angle,  and  let  this  triangle  revolve  on 
SO ;    this  will  form  a  body  which  is  called  an  upright 
cone.     A   sugar  loaf,  a  funnel,  a  carrot 
will   give    a  good   idea  of  a  cone.     The 
side  OA  describes  a  circle  which  is  called 
the  base  of  the  cone.     The  point  S,  which 
has  not  moved,  is  called  the  apex.     The 
length  SO,  the   distance  of   S  from  the 
plane  of  the  base,  is  the  height  of  the  cone. 
Finally,  if  a  circle  turns  round  on  its 
diameter,  the  body  which  this  makes  is 
FIG.  84.          called  a  sphere. 

The  form  of  the  sphere  is  that  of  a 
ball.  The  centre  of  the  circle  is  the  centre  of  the  sphere, 
and  the  radius  of  the  circle  is  the  radius  of  the  sphere. 
Any  plane  which  passes  through  the  centre  cuts  the 
sphere  in  a  circle,  which  has  the  same  radius  as  itself; 
this  circle  is  called  a  great  circle.  Any  straight  line 
which  passes  through  the  centre  cuts  the  sphere  in  two 
points  equally  distant  from  the  centre,  and  the  segment 
limited  by  tnese  two  points  is  a  diameter,  ot  which  the 
length  is  double  that  of  the  radius. 

It  is  well  to  notice  that  a  cylinder  is  defined  when  the 
radius  of  its  base  and  its  height  are  given,  the  same  for  a 
cone,  and  that  a  sphere  is  defined  when  we  know  its 
radius. 

That  being  established,  we  shall  have  the  volume  : — 


SOME  VOLUMES  111 

of  a  cube,  by  multiplying  twice  by  itself  the  length 
of  its  side.  If  a  measures  this  length,  that  gives  us 
a  x  a  x  a  or  a3 :  formula,  V  =  a3 ; 

of  a  parallelepiped,  by  multiplying  the  area  of  the  base 
by  the  height ;  the  area  B  of  the  base  being  itself  a  product 
ab,  if  a  is  a  side  of  the  parallelogram  of  the  base  whose 
height  is  6,  the  product  abh  is  formed  on  multiplying 
by  the  height  of  the  parallelepiped :  formula,  V  =  Eh  = 
abh ; 

of  a  prism,  of  which  the  parallelepiped  is  only  a 
particular  case,  by  multiplying  the  area  of  the  base  by 
the  height :  formula,  V  =  B&  ; 

of  a  pyramid,  by  taking  a  third  of  the  product  of 
the  area  of  the  base  by  the  height ;  this  determination 
of  the  volume  of  the  pyramid  was  given  for  the  first  time 

by  Archimedes1  :   formula,  V  =  —  ; 

of  a  cylinder,  by  multiplying  the  area  of  the  base  by 
the  height.  As  the  base  is  a  circle,  with  radius  r,  if  the 
height  is  h,  it  follows  that  the  formula  is  V  =  •*  r2  h. 

of  a  cone,  by  taking  the  third  of  the  product  of  the  base 

IT  r2  h 
by  the  height :  formula,  V  =  — —  ; 

of  a  sphere,  by  multiplying  the  cube  of  the  radius  by 

4  4 

the  -  of  TT  :  formula,  V  =-^-ir  r3. 

o  o 

It  is  also  established  that  the  area  of  a  sphere  is  equal 
to  four  times  that  of  a  great  circle  or  4  IT  r2.  We  can 
therefore  say  that  the  volume  of  a  sphere  is  equal  to  its 
area  multiplied  by  the  third  of  the  radius. 

Finally,  the  volume  of  a  sphere  can  also  be  determined 

by  the   formula  V  =  g  TT  d3,  and   its  area  by  ird?,  if  we 

call  d  the  diameter. 

All  these  results  will  be  obtained  later.     They  are  only 

1  Archimedes,  an  illustrious  geometrician,  born  at  Syracuse  (287— 
212  B.C.). 


112  MATHEMATICS 

actually  communicated  to  the  pupil  in  order  to  make 
certain  practical  exercises  possible.  But  do  not  ask  him 
to  overload  his  memory  with  all  these  formulae. 

Put  them  afresh  before  his  eyes  every  time  that  he 
needs  them. 

If  by  constant  use  they  should  become  fixed  in  his 
mind,  so  much  the  better.  Otherwise,  pay  no  heed  to  it. 

46.  Graphs;  Algebra  without  Calculations. 

In  many  of  the  reviews  or  journals  of  to-day  we  find 
graphs,  figures  of  which  we  can  make  great  use  for  the 
first  mathematical  education  of  children.  We  must 
make  them  understand  the  signification  of  these  figures, 
and  induce  them  afterwards  to  construct  similar  ones  for 
themselves. 

For  the  most  part,  the  graphs  that  we  are  discussing 
represent  variations  of  meteorological  observations,  for 
instance  barometric  height,  temperature,  or  those  of  the 
market  price  on  the  Stock  Exchange,  over  a  certain  length 
of  time.  We  must  also  remember  that  graphs  are  useful 
in  railway  work,  in  the  representation  of  the  move- 
ment of  trains,  and  that  it  is  the  only  practical  way  of 
keeping  count  of  them. 

But  it  is  above  all  necessary  to  notice  that  by  the  same 
means  we  can  represent  the  variation  of  any  kind  of 
magnitude  which  is  dependent  on  another  magnitude, 
whether  this  is  time  or  anything  else. 

For  example,  when  a  weight  is  hung  on  an  indiarubber 
thread,  this  thread  grows  longer.  It  will  be  possible  to 
make  a  graph  which  would  give  us  the  length  of  the  thread 
if  we  know  the  weight.  When  we  compress  a  gas  its 
volume  diminishes  ;  a  graph  will  tell  us  what  is  the  volume 
of  the  gas  when  we  know  the  pressure.  When  we  heat 
the  steam  from  water  its  pressure  increases ;  a  graph  will 
give  us  the  pressure  if  we  know  the  temperature. 


GRAPHS  113 

In  these  various  examples  the  length  of  the  thread 
depends  upon  the  weight  that  is  suspended;  we  say 
it  is  a  function  of  this  weight ;  the  volume  of  gas 
depends  on  the  pressure ;  it  is  a  function  of  the 
pressure ;  the  pressure  of  steam,  dependent  on  the  tem- 
perature, is  a  function  of  the  temperature.  In  the  pre- 
ceding examples  the  height  of  the  barometer,  the  distance 
traversed  by  a  train,  etc.,  depended  on  the  period,  that 
is,  on  the  time  that  has  elapsed  since  a  certain  fixed 
period.  These  were  functions  of  time. 

This  idea  of  function  is  in  itself  quite  natural,  quite 
simple,  and  a  child  will  easily  grasp  it  if  care  is  exercised 
in  its  mode  of  presentment, 
by  employing  as  many 
examples  as  possible.  When 
a  magnitude  Y  depends  upon 

a  magnitude   X,  and   when  

they  are  both  measurable, 
the  first  is  a  function  of  the 
second. 

The  aim  of  the  graph  is  to  FIG.  85. 

bring  these  functions  before 

the  pupil's  eyes  by  means  of  figures  whose  construc- 
tion is  always  on  the  same  principle.  Let  us  go  into 
this  matter. 

We  will  take  on  squared  paper  (Fig.  85)  two  per- 
pendicular lines  OX,  OY.  To  show  a  particular  value 
x  of  the  variable  quantity  X,  we  will  lay  out  on  OX 
a  length  OP,  which  may  be  measured  by  the  same  number 
as  x,  by  taking  a  certain  unit  of  length.  To  the  value  x 
there  corresponds  a  certain  value  y  of  Y ;  we  represent 
this  by  OQ  on  OY,  taking  whatever  unit  of  length  we 
wish ;  that  done,  we  draw  the  straight  lines  PM,  parallel 
to  OY,  and  QM,  parallel  to  OX,  which  cut  each  other  at 
a  point  M ;  this  point  represents  at  once  the  two  cor- 
responding values  x,  y.  By  thus  constructing  points, 
like  M,  as  many  as  we  like,  and  joining  them  by  means  of 

M.  I 


114  MATHEMATICS 

a  continuous  line,  the  graph  showing  the  variation  of 
the  function  Y  is  obtained. 

If  the  quantity  X  has  negative  values,  and  x  be  one  of 
them,  the  point  P,  instead  of  being  to  the  right  of  O,  will 
be  to  the  left  on  OX. 

If  the  quantity  Y  has  negative  values,  and  y  be  one  of 
these,  the  point  Q,  instead  of  being  above  O,  will  be 
below,  on  OY. 

For  any  two  values  whatever  which  correspond,  that 
is,  represent  both  at  once  the  two  points  P,  Q,  there 
is  always  one  point  M  and  no  more. 

If  the  points  M  which  we  obtain  are  not  very  close 
together,  we  can  join  them  by  segments  of  straight 
lines ;  we  do  not  even  try  to  picture  by  a  curve  the 
function  Y ;  but  the  points  which  show  this  outline  in 
segments  of  straight  lines  will,  all  the  same,  give  a  general 
idea  of  the  manner  in  which  this  function  varies. 

In  algebra — as  we  shall  see  later — we  do  very  little  but 
study  the  functions  which  can  be  determined  by  calcula- 
tion, and  which  are  called,  for  this  reason,  algebraical 
functions ;  and  the  fundamental  problem  of  algebra 
consists  in  finding  values  of  X,  such  that  two  functions 
Y,  Z,  of  X,  become  equal  to  one  another.  We  see 
(Fig.  85)  that,  if  the  two  graphs  (Y),  (Z),  of  the  functions 
Y,  Z  were  traced,  these  lines  would  cut  each  other  at 
the  points  MI,  M2 ;  by  taking  MjP^  M2P2,  parallel  to 
OY,  as  far  as  OX,  the  lengths  OPl5  OP2  will  then  give, 
with  the  unit  adopted  to  measure  the  lengths  on  OX, 
the  two  numbers  #1}  x.2  which  it  was  required  to  find. 

It  is  in  this  sense  that  we  can  see  that  graphs  allow  us 
to  work  bv  algebra  without  calculations,  and  even  more 
than  that,  since  it  has  been  possible  to  establish  graphs 
for  functions  which  are  not  algebraical.  We  ought  to 
add  that  all  the  results  thus  obtained  are  not  rigorously 
exact.  Howrever,  in  practice,  in  a  great  number  of  cases, 
if  the  graphs  are  carefully  made,  this  approximate  result 
will  be  all  that  is  necessary.  There  are  many  questions 


THE  TWO  WALKERS 


115 


to  solve  which  these  outlines  may  be  applied  with  advan- 
tage ;  besides,  they  speak  to  the  mind  through  the 
intermediary  of  the  eyes,  and  absolutely  place  a  living 
representation  before  the  pupil.  This  is,  in  itself,  a 
valuable  aid  to  the  teacher. 

In  the  following  sections  some  examples  will  serve 
still  further  to  enlighten  the  child's  mind  as  to  the  con- 
struction and  the  employment  of  graphs.  Their  most 
natural  application  seems  to  be  in  solving  the  type  of 
problem  known  under  the  name  of  travelling  problems  ; 
thus  we  shall  especially  work  with  these  in  various  forms. 

47.  The  Two  Walkers. 

Here  we  give,  under  one  of  its  most  simple  forms,  an 
example  to  show  what  the  travelling  problem  is.  A 
pedestrian  starts  at  a  given  hour,  Y 
from  a  given  place,  at  a  certain 
known  speed.  Some  time  after- 
wards, a  second,  going  at  a  greater 
speed,  starts  out  in  the  same  direc- 
tion, following  the  same  route. 
When  will  he  overtake  the  first,  and 
at  what  distance  from  his  point 
of  departure  ? 

To  solve  this  problem,  and  others  of  the  same  kind,  we 
must  see  how  the  graph  of  a  pedestrian  is  made.  To 
do  this,  on  a  piece  of  squared  paper  (Fig.  86)  let  us  take 
our  two  perpendicular  straight  lines  OT  (on  which  we 
shall  mark  the  time)  and  OY  (on  which  we  shall  mark 
the  distances).  The  point  O  corresponds  to  midday, 
for  instance ;  let  2  divisions  mark  an  hour,  and  mark 
along  OT,  lh.,  2h.,  3h.  Again  on  OY,  starting  from 
O,  let  a  division  represent  a  mile,  and  mark  1m.,  2m., 
3m.  ...  If  a  man  starts  at  half-past  two,  with  a  speed  of 
3  miles  an  hour,  it  will  be  seen,  to  begin  with,  that  the 
graph  will  contain  the  point  A  on  OT;  then  that  at 

12 


5m 
4m 
3m 
2m 
1  m 


lh 


2hA 
FIG.  86. 


116  MATHEMATICS 

half-past  three  he  will  have  gone  3  miles,  which  gives  the 
point  B  ;  finally,  as  the  man  goes  on  at  3  miles  an  hour 
regularly,  the  straight  line  AB  will  be  the  required  graph. 
We  see  that  at  four  o'clock  he  will  be  4£  miles  from  his 
starting  point,  and  that  the  simple  outline  of  the  straight 
line  AB  shows  us  at  what  distance  the  man  finds  himself 
at  a  pre-determined  hour,  and  at  what  hour  he  has  gone 
over  a  given  distance. 

We  will  come  back  now  to  our  question,  and  make  it 
more  precise  by  saying  that  a  child  starts  with  a  speed 
of  2  miles  an  hour,  and  that  a  man  starts  1  hour  after  him 
at  a  speed  of  3  miles  an  hour.  Taking  (Fig.  87)  the  same 

units  as  just  now,  and  counting 
the  time  of  starting  from  the 
departure  of  the  child,  then  the 
straight  line  OBj  is  the  graph  of 
the  child,  and  A.2B2  is  the  graph 


7m 
6m 
5m 
4m 
3m 
2m 
I  m 


of  the  man. 

These   lines   cut    each    other 

Ih     2h    3h     *h      T     a^  M»  corresponding  to  3  hours 
F      g7  and  6  miles.     The  meeting  will 

then  take  place  at  6  miles  from 

the  point  of  departure,  and  3  hours  after  the  start  of  the 
child. 

The  problem  can  now  be  completed  by  complicating 
it  a  little.  At  a  place  7  miles  from  the  starting  point 
a  carriage  is  sent  out,  going  before  the  two  travellers 
but  in  the  opposite  direction.  The  carriage  starts  half- 
an-hour  after  the  child,  at  the  rate  of  4  miles  an  hour. 
Where  will  it  meet  each  of  the  two,  and  at  what  time  ? 
A3B3  is  the  graph  of  the  carriage.  This  straight  line  cuts 
OB!  at  the  point  B3,  showing  1^  hours  and  3  miles;  that 
gives  the  time  and  place  of  meeting  with  the  child.  The 
meeting  place  with  A2B2  is  at  about  Ij  hours  (rather  less), 
and  at  rather  more  than  2j  miles. 

Treating  this  question  by  ordinary  calculation  we  would 
arrive  at  1  hour  43  minutes  as  the  time,  and  2}  miles  as  the 


FROM  PARIS    TO   MARSEILLES 


117 


distance.  It  is  easy  to  see,  despite  the  small  dimensions 
of  Fig.  87,  that  it  places  results  before  our  eyes 
almost  exactly  correct  and  perfectly  satisfactory  in 
practice. 


48.  From  Paris  to  Marseilles. 

In  the  table  of  trains  between  Paris  and  Marseilles, 
we  have,  from  the  beginning  of  the  year  1905,  chosen 
the  express  train  No.  1  from  Paris  to  Marseilles  and 


Midday. 
»«».)«     1            If}* 

0   II 

Midnight. 
S      «       ?     t      «      10     «.    - 

J 

JO  ±\ 

WO.             i   4- 

150  ...          \ 

/J 

I5i                   i<n'« 
MB                               \ 

> 

HO                                  v                  -    -L  -  H 

/ 

300  _,_  v  

''i   •' 

sis  '"                      "  •;•" 

i50                                                \ 

/ 

WO  J_          J.                                    * 

yf    i 

/ 

500  ... 

J 

512  T1                                              "••  " 
550                                                        t 

600  (  

' 

650  j_                                         j 

BB  .  .              -(-'  j-+  

T     i  v^            T 

BOO             _|_              /                     T 

1  1 

F  11    i      ml 

tlllltl     1           1 

FIG.  88. 


No.  16  from  Marseilles  to  Paris  (both  day  trains)  to  make 
the  same  figure  (Fig.  88)  show  graphs  of  them.  The 
time  is  marked  horizontally,  the  distance  (in  kilometres) 
vertically.  The  horizontal  lines  at  odd  distances  repre- 


118 


MATHEMATICS 


sent  various  places  at  various  (vertical)  distances  from 
either  end.  On  these  lines  are  marked  both  the  time  of 
arrival  and  of  departure.  The  two  graphs  cut  at  a  point 
which  tells  us  when  the  two  trains  meet. 

We  give  below  the  tables  showing  the  hours  at  which 
the  different  stations  on  the  two  journeys  are  reached, 
in  order  that  a  comparison  may  be  made  between  the 
variations  of  the  journey  and  the  absolute  facts — and 
this  despite  the  limited  dimensions  of  the  figure. 


TRAIN  No.  1. 


TRAIN  No.  16. 


- 

Arrival. 

Departure. 

- 

Arrival. 

Departure. 

Paris 

9.20  a.m. 

Marseilles 

11.53  a.m. 

Laroche 

11.34 

11.39 

Avignon 

1.20 

1.26  p.m. 

Dijon 

1.59 

2.15  p.m. 

Valence 

2.51 

2.54 

Macon 

0.->1 

3.54 

Lyons 

4.8 

4.14 

Lyons 

4.57 

5.14 

Macon 

5.10 

5.13 

Valence 

6.39 

6.42 

Dijon 

6.40 

6.46 

Avignon 

8.16 

G.27 

Laroche 

8.31 

8.36 

Tarascon 

8.45 

8.53 

Paris 

10.20  p.m. 

— 

Marseilles 

10.11  p.m. 

We  will  take  advantage  of  this  occasion  to  notice  what 
a  desirable  thing  it  would  be  if,  at  any  rate  as  far  as  the 
railway  is  concerned,  the  habit  could  be  formed  of  reckon- 
ing the  hours,  starting  from  midnight,  from  0  to  24.  This 
would  avoid  the  use  of  the  words  morning  and  evening, 
which  prove  a  fruitful  source  of  mistakes  and  cause  endless 
confusion.  In  some  civilised  countries,  this  is  already 
done,  but  not  yet  in  the  United  States  ;  let  us  hope  that 
some  centuries  hence  people  will  succeed  in  understanding 
that  it  is  quite  as  easy  to  say  "  17  o'clock  "  as  5  p.m. 

I  cannot  sufficiently  impress  upon  the  student  the  im- 
portance of  making  use  of  railway  guides  in  the  con- 
struction of  graphs  of  this  nature,  and  choosing  purposely 
places  c:ose  at  hand,  localities  which  they  know  already, 


FROM   HAVRE   TO  NEW  YORK  119 

at  least  by  name.  Little  time  will  be  lost  if  squared  paper 
is  used  and  the  outlines  done  in  freehand.  They  will 
then  prove  very  useful  exercises. 

Single-gauge  lines  afford  matter  for  most  interesting 
remarks  on  the  outline  of  graphs,  to  show  the  crossing 
of  trains  which  are  running  in  opposite  directions. 

There  is  opportunity  to  notice  also  how  one  train  going 
at  a  greater  speed  than  another  is  able  to  pass  it,  the 
slower  one  switching  into  a  station,  where  it  remains  to 
give  the  quicker  train  sufficient  time  to  run  in  front. 
There  really  can  be  no  indication  of  the  thousand  inter- 
esting details  which  the  construction  and  observation  of 
these  graphs  suggest  to  us. 


49.  From  Havre  to  New  York. 

A  long  time  ago,  during  a  scientific  congress,  a  number 
of  well-known  mathematicians  of  various  nationalities — 
some  being  of  world-wide  reputation — were  dining  to- 

Havre  0  1  ?  3  *  5  6   78  9  10 


BewYork 

0123456789  10  U  12  13  14  IS  16  17 

a 

FIG.  $'.). 


gether.  At  the  end  of  the  meal  Edward  Lucas  suddenly 
announced  that  he  was  going  to  lay  before  them  a  most 
difficult  problem. 


120  MATHEMATICS 

"  Suppose,"  said  he,  "  (it  is  unfortunately  only  a 
supposition)  that  each  day,  at  midday,  a  packet  boat 
starts  from  Havre  to  New  York,  and  that  at  the  same  time 
a  similar  boat,  belonging  to  the  same  company,  leaves 
New  York  for  Havre.  The  crossing  is  made  in  exactly 
seven  days,  in  either  direction.  How  many  of  the  boats 
of  the  same  company  going  in  the  opposite  direction  will 
the  packet  boat  starting  from  Havre  to-day  at  midday 
meet  ?  " 

Some  of  his  distinguished  hearers  foolishly  answered 
"  seven."  The  greater  number  kept  silence,  appearing 
surprised.  Not  one  gave  the  exact  solution  which  appears 
with  perfect  clearness  on  Fig.  89. 

This  anecdote,  which  is  absolutely  true,  instructs  us 
in  two  ways.  To  begin  with,  how  patient  and  lenient  we 
ought  to  be  with  those  children  who  cannot  immediately 
take  in  things  which  are  entirely  strange  to  them.  Then, 
again,  the  question  asked  by  Lucas  shows  us  the  extreme 
usefulness  of  graphs  as  the  best  method  of  solving  similar 
problems.  Really,  if  the  most  ordinary  of  the  mathe- 
maticians had  had  this  idea,  Fig.  89  would  have 
arranged  itself  in  his  brain  ;  he  would  have  seen  it,  as  it 
were,  with  his  mind's  eye,  and  would  not  have  hesitated. 
But  they,  on  the  contrary,  only  thought  of  the  ships 
about  to  start,  and  forgot  those  on  the  way — reasoning 
but  not  seeing. 

It  is  certain  that  any  boat  of  which  the  graph  is  AB 
will  meet  at  sea  13  other  boats  of  the  fleet,  plus  the  one 
entering  Havre  at  the  moment  of  departure,  plus  the  one 
leaving  New  York  at  the  moment  of  arrival — 15  in  all.  At 
the  same  time,  the  graph  shows  the  time  of  meeting  to 
be  midday  and  midnight  of  each  day. 

To  Lucas  also  we  owe  the  problem  to  be  found  in  his 
Arithmetique  Amusante  under  the  name  of  "  The  Ballad 
of  the  Slipping  Snail,"  formulated  thus : 

"  A  snail  begins  to  climb  up  a  tree  one  Sunday  morning 
at  six  o'clock ;  during  the  day,  up  to  six  o'clock  in  the 


WHAT   KIND   OF  WEATHER   IT  IS        121 

evening,  he  gets  up  5  yards  ;  but  during  the  night,  he 
falls  back  2  yards.  At  what  time  will  he  have  climbed  up 
9  yards  ?  " 

This  is  again  a  travelling  problem  (slow  travelling  this 
time  !).  A  bewildered  child  will  answer,  "  Wednesday 
morning,"  which  is  wrong.  I  leave  to  my  readers  and 
their  pupils  the  pleasure  of  discovering  the  answer  by 
tracing  the  graph  of  the  "  scramble  of  the  slipping  snail." 


50.  What  kind  of  Weather  it  is. 

Here  we  give  (Fig.  90)  two  graphs  at  once,  one  dealing 
with  barometric  pressure,  the  other  relating  to  tempera- 
ture, during  the  last  week  of  the  year  1881.  We  are 
borrowing  these  from  a  journal  ("  La  Nature  "),  but  taking 
away,  for  simplicity's  sake,  several  of  the  other  things 
shown  therein. 

Here,  we  only  wish  to  show  how  variations  of  functions, 
about  which  we  have  nothing  to  go  upon  but  experience, 
are  suitably  shown  by  the  graph  method. 

It  also  seems  interesting  to  indicate  how  two  different 
functions  can  be  shown  at  once  on  one  figure  with  perfect 
clearness. 

The  line  which  indicates  the  variations  of  the  barometer 
is  drawn  heavily,  while  the  thermometer  variations  are 
shown  by  a  dotted  line. 

To  read  the  barometric  pressures,  look  to  the  left 
of  the  figure,  while  the  figures  to  show  variations  of  the 
thermometer  (in  degrees  Centigrade)  are  to  be  found  on 
the  right. 

No  confusion  is  possible  at  all.  These  graphs  have 
rendered  the  greatest  service  in  Meteorology,  and  have 
largely  contributed  to  spread  a  knowledge  of  this  science, 
so  useful  even  now,  which,  although  only  in  its  in- 
fancy, is  progressing  by  leaps  and  bounds. 


122 


MATHEMATICS 


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§ 


TWO   CYCLISTS   FOR   ONE    MACHINE      123 


miles 
20 

is 

10 


51.  Two  Cyclists  for  One  Machine. 

Two  cyclists  having  arranged  a  certain  journey,  one 
of  them  unfortunately  found  himself  obliged  to  wait 
until  some  necessary  repairs  could  be  done  to  his  cycle. 
However,  they  decided  not  to  delay  their  journey,  so  they 
arranged  it  in  the  following  way  :  that  they  should  start 
together,  one  on  the  cycle,  the  other  on  foot ;  that  at 
a  certain  point  the  cyclist  should  deposit  his  machine 
in  a  ditch  at  the  side  of  the 
road  and  continue  his  journey 
on  foot.  His  companion,  on 
arrival  at  the  spot  agreed  upon, 
should  then  mount  the  machine, 
and  rejoin  the  other,  when  the 
same  thing  would  be  repeated. 

The  programme  as  arranged 
was  duly  carried  out,  and  the 
last  day  found  them  with 
20  miles  yet  to  go.  When 
cycling,  each  traveller  goes 

7\  miles  an  hour;   when  walking,  the  speed  of  each  is  2| 
an  hour. 

At  what  point  ought  the  bicycle  to  be  put  on  one  side 
by  the  first  traveller  (no  more  changing  taking  place)  so 
that  both  may  arrive  at  their  journey's  end  at  the  same 
time? 

The  answer  is  evident.  As  each  has  to  go  the  same 
distance  on  foot,  as  also  by  cycle,  the  last  change,  in 
order  to  arrive  at  their  destination  at  the  same  time, 
should  take  place  half-way,  that  is,  10  miles  from  the 
starting  point. 

Fig.  91  gives  us  the  graphs  of  the  journeys  of  the 
two  travellers,  the  first  shown  by  a  continuous  line,  the 
second  by  a  dotted  one.  To  make  the  explanation  easier, 


5h 


FIG.  91. 


124  MATHEMATICS 

we  will  suppose  that  the  departure  takes  place  at  midday. 
The  cyclist  arrives  half-way  at  20  minutes  past  1  ; 
there  he  leaves  the  machine  and  goes  forward  on  foot ; 
he  finishes  his  10  miles,  which,  at  2|  miles  an  hour,  will 
bring  him  to  the  end  of  his  journey  at  20  minutes  past  5. 

His  friend,  setting  out  on  foot,  does  his  first  10  miles 
by  4  o'clock,  then  he  mounts  the  cycle,  and  also  arrives 
at  20  minutes  past  5. 

In  short,  they  have  5  hours  and  20  minutes  to  do 
20  miles,  which  averages  3|  miles  an  hour.  We  can  see 
that  this  mode  of  locomotion  adds  sensibly  to  the  speed 
of  a  pedestrian ;  it  may  be  worth  something  as  a  hint  to 
two  young  men  who  may  have  just  enough  money  to 
buy  one  machine  and  can  share  the  advantage  of  it  in 
this  manner. 

To  make  this  arrangement  workable  in  practice,  the 
changes  would  have  to  be  made  pretty  frequently,  so 
that  the  cycle  would  not  be  left  long  without  a  rider 
(unless,  of  course,  the  country  through  which  the  journey 
was  made  was  either  very  deserted,  or  the  people  of 
exceptional  honesty).  On  Fig.  91  we  have  indicated 
this  variation ;  supposing  that  the  machine  is  aban- 
doned at  5  miles,  then  at  15  from  the  point  of  departure 
OoaM  would  be  the  graph  of  one  of  the  travellers, 
and  O66M  would  be  that  of  the  other.  Here  the  friends 
would  join  half-way,  but  the  cyclist  would  go  on,  leaving 
his  companion  walking.  Graphs  take  into  account  all 
these  circumstances. 

They  would  apply  equally  to  two  travellers  not 
possessing  the  same  average  speed,  whether  as  pedestrians 
or  cyclists.  We  can  thus  work  problems  which  lend 
themselves  to  calculation  without  any  great  difficulty, 
but  which  require  a  knowledge  of  Mathematics, 
which  we  do  not  suppose  the  children  possess  even  in 
the  smallest  possible  degree. 


THE  CARRIAGE  THAT  WAS  TOO  SMALL     125 


miles 

Y  31)4 
1  30 


P25& 


10 

Q  6 


M 


52.  The  Carriage  that  was  too  Small. 

Four  travellers  (Mr.  and  Mrs.  Tompkins  and  Mr.  and 
Mrs.  Wilkins)  arrived  one  morning  at  the  station  of 
X  .  .  .,  intending  to  go  for  the  day  to  Y  .  .  .,  a  little 
village  about  31-.\-  miles  distant,  which  they  proposed 
reaching  in  time  for  dinner.  They  had  been  told  that 
they  would  be  able  to  hire  a  motor  on  arrival  which  woul<  I 
quickly  take  them  to  their  hotel,  or  wherever  they  were 
going  to  dine,  along  a 
delightful  road.  The  infor- 
mation proved  to  be  correct, 
but  unfortunately  the  only 
available  motor  would  only 
hold  two  people  and  the 
chauffeur.  Its  speed  was 
15  miles  an  hour. 

My  readers  can  picture 
the  situation.  None  of  the 
four  prided  themselves  on 
their  powers  as  pedestrians  ; 
they  were  old  and  liked  to 

do  their  modest  2  miles  per        ^  2      3      4      5h 

hour,  but  no  more. 

However,  it  was  settled 
that  the  Tompkins  should 

start  in  the  motor  and  that  the  Wilkins  should,  at  the 
same  time,  set  out  on  foot.  At  a  certain  distance  the 
motor  would  put  down  the  T.'s,  who  would  proceed  on 
foot,  go  back  to  pick  up  the  W.'s,  and  carry  them  to 
their  destination.  How  would  this  be  managed  so  that 
all  of  them  would  arrive  at  the  same  time,  and  how  long 
would  it  take  to  make  the  journey  ? 

These  questions  are  not  very  puzzling,  but  it  is  a  good 
thing  to  be  able  to  solve  them. 


FIG.  92. 


126  MATHEMATICS 

This  problem,  except  for  insignificant  changes  in  the 
data,  has  been  given  at  some  competitive  examinations. 

It  bears  a  certain  analogy  to  that  of  the  last  section, 
but  is  slightly  more  complicated,  owing  to  the  fact  that 
the  motor  has  to  come  back  to  pick  up  Mr.  and 
Mrs.  Wilkins. 

If  we  show  by  P  the  place  on  the  route  where  the 
Tompkins  leave  the  vehicle,  by  Q  the  point  where  it 
takes  up  the  Wilkins,  the  four  points  X,  Q,  P,  Y  are 
arranged  in  this  order :  the  Tompkins  go  from  X  to  P 
by  motor,  from  P  to  Y  on  foot ;  the  Wilkins  go  from 
X  to  Q  on  foot,  from  Q  to  Y  by  motor.  So  that  they 
may  all  arrive  together  it  follows  that  XP  =  QY  and 
XQ  =  PY,  which  comes  to  the  same  thing ;  and  con- 
sequently, as  just  now,  the  graph  of  the  journey  of  the 
T.'s,  and  that  of  the  W.'s  will  form  a  parallelogram 
(Fig.  92).  But  whilst  in  Fig.  91  the  diagonal  of  this 
parallelogram  was  parallel  to  the  axis  on  which  time 
is  measured,  the  cycle  remaining  at  rest  in  the  ditch, 
here  it  will  be  totally  different.  The  diagonal  CD  will 
be  no  other  than  the  graph  of  the  motor  journey  when 
it  comes  back  part  way  for  the  W's. 

On  Fig.  92,  XCM  shows  the  journey  of  the  T.'s, 
XDM  that  of  the  W.'s,  and  XCMD  is  a  parallelogram. 
These  remarks  furnish  the  means  whereby  the  figure 
can  very  easily  be  constructed.  To  begin  with,  it  is 
sufficient  to  draw  the  two  straight  lines  XC  and  XD, 
which  is  quite  simple,  since  we  have  the  speed  of  the 
motor  (15  miles  an  hour)  and  that  of  the  pedestrians 
(2  miles  an  hour).  Taking  then  a  point  d,  anywhere  on 
XC  (suppose  we  say  the  one  which  agrees  with  1  hour 
and  15  miles),  we  draw  CJ)],  which  represents  the 
graph  of  the  return  of  the  motor  if  it  comes  back  again 
to  start  from  Ct ;  this  straight  line  cuts  at  Dj  the  straight 
line  XD.  Let  us  take  the  middle  Ot  of  CJX,  and,  joining 
X  and  O1}  the  straight  line  XO],  produced  to  the  point 
M,  which  corresponds  to  a  distance  of  31  \  miles,  will 


THE  DOG  AND  THE  TWO  TRAVELLERS  127 

give  us  the  extremity  M  of  the  two  graphs :  draw 
MC  parallel  to  XD,  MD  parallel  to  XC  ;  the  parallelogram 
will  be  completely  drawn,  and  XCDM  will  represent  the 
graph  of  the  motor.  Then  we  see  on  the  figure  that  D 
corresponds  to  3  hours  and  6  miles,  C  to  about  If  hours 
and  25|  miles,  M  to  4f  hours,  and,  naturally,  31 1  miles. 

Therefore  the  motor  ought  to  put  down  the  T.'s  at 
a  distance  of  25  J  miles  at  about  1.45  ;  then  come  back 
to  pick  up  the  W.'s,  finding  them  at  3  o'clock,  6  miles 
from  the  starting  point,  and  bring  them  on  to  meet  the 
T.'s  at  4.45.  An  exact  calculation  would  make  the  arrival 
4.42  instead  of  4.45,  but  this,  in  practice,  is  of  very  slight 
importance. 

Summing  up  the  problem,  the  travellers  ought  to  travel 
25|  miles  by  motor  and  6  on  foot,  and  the  entire  journey 
is  effected  in  4  hours  42  minutes. 

The  average  speed  is  about  6*7  miles  an  hour,  meaning 
that  they  arrive  at  their  destination  at  the  same  time  as 
if  all  the  journey  had  been  made  in  a  motor  with  a  speed 
of  6'7  miles  an  hour.  The  travellers — both  the  T.'s  and  the 
W.'s — walking  3  hours,  would  have  done  6  miles,  and  the 
remainder  by  motor ;  as  for  the  motor,  it  would  have 
run  in  all  70|  miles,  as  follows  :  25|  onward,  19|  back- 
ward, and  again  forward  for  25 1  more. 

This  example,  treated  thus  in  detail,  will  serve  as  a 
theme  for  numerous  similar  exercises,  making  use  of 
different  data. 


53.  The  Dog  and  the  Two  Travellers. 

Two  travellers  are  going  along  a  road  in  the  same 
direction.  The  first,  A,  is  6  miles  in  front  of  the  other, 
and  walks  3  miles  an  hour  ;  the  second,  B,  walks  4|  miles 
an  hour.  One  of  the  travellers  has  a  greyhound,  who, 
at  the  exact  moment  of  which  we  speak,  runs  to  the  other 
at  a  speed  of  ll£  miles  an  hour,  running  immediately 


128 


MATHEMATICS 


18 


back  to  his  master.  Having  rejoined  him,  he  starts  off 
to  do  the  same  thing  again,  and  continues  this  until  the 
men  meet,  zigzagging  from  one  to  the  other.  What  is 
wanted  is  the  distance  the  dog  will  have  travelled  up 
to  the  moment  of  meeting. 

It  appears  that  the  question  can  be  put  in  two  ways, 
according  to  which  of  the  men  is  the  dog's  owner.  In 
Fig.  93  the  time  is  counted  from  the  moment  the  dog 
is  let  loose.  The  graphs  of  the  two  travellers  are  OM 
6M,  and  and  the  point  M,  which  represents  the  meeting, 

corresponds  to  18  miles  and 
4  hours  of  walking.  If  the 
dog  belongs  to  the  traveller 
who  is  at  the  back,  his  graph 
is  Qaa  .  .  .,  a  line  taking  a 
zigzag  course  between  the 
journeys  of  the  two  men.  If, 
on  the  contrary,  the  animal 
is  the  property  of  the  man 
in  front,  his  graph  is  Qbb  .  .  ., 
a  line  of  the  same  nature  but 
different  from  the  first.  In 
any  case,  the  dog  has  never 
ceased  running  for  4  hours, 
and  as  he  goes  at  a  speed  of 

11  \  miles,  he  will  consequently  have  covered  a  distance  of 
45  miles.  Whichever  hypothesis  we  take,  the  result  will 
be  the  same. 

We  have  taken  exceptionally  simple  instances,  to 
make  the  explanations  very  easy.  It  will  be  useful  to 
vary  them  in  the  exercises  which  may  be  given  on  this 
subject ;  for  instance,  we  might  suppose  that  the  men 
start  in  opposite  directions,  advancing  till  they  meet. 

54.  The  Falling  Stone. 

In  the  travelling  graphs  that  we  have  seen  up  to  the 
present  time,  whether  we  deal  with  pedestrians,  carriages, 


3 
Vk. 


234 
FIG.  93. 


5  h 


THE   FALLING  STONE 


129 


railways,  or  dogs,  the  distance  passed  over  in  a  given 
time,  in  a  second,  say,  was  always  the  same,  and  it 
followed  that  the  graph  was  a  straight  line.  That  is 
explained  by  saying  that  the  speed  was  constant,  or  that 
the  movement  was  uniform. 

It  is  not  at  all  the  same  for  a  stone  that  is  thrown  to 
a  certain  height  and  then  is  allowed  to  drop.  Experience 
teaches  us,  if  we  do  not  take  into  account  the  resistance 
of  the  air,  that,  at  the  end  of  a  second,  the  stone  will  have 
fallen  nearly  16  feet.  At  the  end  of  2  seconds  it  will 
have  fallen  64  feet,  and  at  the  end  of  the  third  144  feet. 
This  shows  us  that  the  graph  of  this  movement  (Fig.  94) 
will  take  the  form  of  a  curve,  and  no  longer  that  of  a 
straight  line.  This  curve  will 
be  pretty  nearly  the  one  in- 
dicated by  the  figure.  It  is  a 
fragment  of  a  line  about  which 
we  shall  speak  further  in  a 
little  while,  and  is  called  a 
parabola. 

Writing  the  formula  y  = 
16£2,  we  have  the  distance  y 
travelled  by  the  stone  in  its 
fall  when  it  has  fallen  during 
a  certain  time  f,  provided  that,  FlG-  94> 

in  measuring  the  time,  we  take  the  second  as  the  unit  ; 
then  the  number  y  which  will  be  obtained  will  be  a 
number  of  feet. 

For  example,  in  -j^th  of  a  second  the  stone  only  falls 
2  inches,  and,  as  we  have  just  said,  at  the  end  of  a 
second,  it  has  fallen  16  feet.  In  10  seconds  it  will  have 
travelled  1,600  feet.  Thus  we  see  that  it  falls  quicker 
and  quicker ;  in  other  words,  its  movement  becomes 
accelerated. 

To  fall  from  a  height  of  .900  feet  would  take  a 
stone  about  1\  seconds,  always  supposing  that  we  do 
not  take  into  account  the  resistance  of  the  air.  In 

M.  K 


64- 


144 

Y 


130  MATHEMATICS 

practice  this  is  only  very  little  when  we  are  consider- 
ing little  distances,  but  it  becomes  very  appreciable 
when  great  heights  are  in  question,  and  it  is  a  mistake 
to  think  that  our  graph  will  then  be  a  correct  representa- 
tion. 

55.  The  Ball  Tossed  Up. 

If  we  toss  a  leaden  ball  into  the  air  it  will  rise  to  a 
certain  height,  then  fall  down.  Following  the  object 
with  a  certain  amount  of  attention,  it  is  not  difficult 
to  prove  that  the  movement  becomes  slower  and  slower 
during  the  ascent,  while  during  the  descent,  on  the 
contrary,  the  motion  becomes  quicker.  In  the  first 
period  the  movement  is  slackened,  in  the  second  it  is 
accelerated. 

Instead  of  using  the  hand,  let  us  suppose  that  we 
employ  a  gun,  the  barrel  of  which  is  placed  vertically; 
the  same  effect  would  be  noticed  ;  only  we  must  remember 
that  the  greater  the  speed  at  which  the  ball  is  launched 
the  more  it  will  rise,  and  the  more  time  will  elapse  before 
it  falls  to  earth. 

It  is  interesting  to  find  out  various  details  about 
the  movement,  to  know,  especially,  to  what  height  the 
ball  will  rise  ;  how  long  it  will  take  to  get  to  this  height ; 
how  lon^  it  will  take  in  its  descent. 

When  we  know  the  speed  a  at  which  the  ball  has  been 
thrown,  and  which  is  known  as  the  initial  velocity,  all 
the  answers  to  these  questions  are  given  by  the  formula 
y  =  at  -  IGt2. 

To  comprehend  its  meaning,  and  to  make  use  of  it 
when  necessary,  we  must  know  : 

1.  That  the  height  y  is  measured  in  feet ; 

2.  That   the   initial    velocity   a   is   measured   in    feet 
per  second  ;    that  is  to  say,  that  the  ball  is  thrown  in 
such  a   manner  that  if   nothing  caused   a  slackening  of 


THE  BALL  TOSSED  UP  131 

speed,  it  would  go  on  indefinitely  travelling  a  feet  each 
second ; 

3.  That  the  time  /  is  measured  in  seconds. 

However  simple  the  calculations  may  be  to  which  this 
formula  leads,  we  can  follow  the  movement  more  easily 
by  means  of  a  graph  (Fig.  95). 

It  has  been  constructed  on  the  supposition  that  a  =  64, 
that  is  to  say,  that  the  ball  is  thrown  in  such  a  way  that 
it  would  travel  64  feet  per  second  if  nothing  happened 
to  oppose  its  movement.  If  we  construct  the  straight 
line  OA,  which  would  be  the  graph  of  this  uniform  move- 
ment of  64  feet  a  second,  there  is  a  very  simple  means  of 
obtaining  what  we  wish  ;  going  back  to  Fig.  94,  we  set 
out  exactly  the  same  heights  for  1  second,  2  seconds, 
3  seconds,  etc.,  but  below  OA  (Fig.  95)  instead  of  being 
below  OT  (Fig.  94). 

Another  method  is  to  make 
use  of  the  formula  above, 
at  --  IGt2,  in  order  to  have 
each  value  of  y. 

By  using  any  of  these,  we 
shall  see,  on  the  hypothesis 
that  a  =  64,  that  the  ball  will 
rise  for  2  seconds,  that  it  will 
reach  a  height  of  64  feet,  and  that  it  will  take  2  seconds 
to  come  down.  The  line  obtained  has  again  the  form  of 
a  parabola. 

In  a  general  way  it  is  found  that  the  time  taken  up  by 
the  descent  will  be  always  the  same  as  that  of  the  ascent, 
and  that  the  height  to  which  the  ball  will  reach  is  always 

a- 

— '  expressed  in  feet. 

Here,  as  in  the  preceding  section,  it  is  quite  understood 
that  no  account  is  taken  of  the  resistance  of  the  air, 
which,  however,  for  big  initial  speeds,  would  have  a 
sensible  effect,  in  both  going  up  and  coming  down. 


K  2 


182 


MATHEMATICS 


56.  Underground  Trains. 

Underground,  or  tube,  railway  systems  show  special 
working  conditions,  made  necessary  by  the  needs  of 
travelling  service  in  a  big  city. 

To  begin  with,  the  stations  are  very  close  together ; 
often  only  some  hundred  yards  lie  between  them.  Besides 
that,  the  trains  follow  each  other  at  short  intervals,  so 
that  the  stop  at  each  station  has  to  take  as  little  time  as 
possible. 

Under  such  conditions  a  good  part  of  the  time  needed 
for  the  journey  from  one  station  to  another  is  employed, 

on  leaving  one  stopping 
place,    in     quickening 
M  the     speed ;     then,     on 

approaching  the  next, 
in  slackening  it  off  ;  this 
latter  is  done  by  means 
of  brakes,  for  if  the 
train  were  brought  to  a 

standstill    suddenly,    an 

20     30     40    50    60s    T   accident  might  happen. 
jrIG>  96.  Our  readers  might  say 

that  this   holds   for   all 

railway  trains,  which  is  partly  true  ;  but  as  the  distances 
between  two  stations  are  sufficiently  long,  the  periods  of 
setting  the  train  in  motion  and  applying  the  brake  count 
for  very  little  in  the  whole.  This  is  why,  without  depart- 
ing from  practical  exactitude,  we  can  represent  by  a 
straight  line  the  graph  showing  the  journey  of  a  train 
between  two  stations. 

This  journey  is,  then,  interesting,  because  of  these 
peculiarities,  and  also  of  the  corresponding  graph,  which 
is  shown  in  Fig.  96. 

To  draw  this  graph  we  have  supposed  two  stations 
distant  400  yards  one  from  the  other,  a  speed  through 
the  entire  journey  of  36,000  yards  (about  20  miles)  an 


300 
200 


100 


ANALYTICAL   GEOMETRY  133 

hour,  which  means  10  yards  a  second  ;  finally  it  must  be 
admitted  that  it  takes  20  seconds,  starting  from  the 
halt,  to  get  up  full  speed ;  and  equally,  of  course,  20 
seconds  to  slacken  off  before  the  next  stopping  place. 

With  these  data  to  hand,  corresponding  to  the  working 
of  the  journeys  we  see  that  a  train  starting  to  the 
next  station  moves  at  a  rising  speed,  like  a  ball  which  falls 
faster  and  faster ;  it  runs  over  100  yards  in  20  seconds  ; 
it  rolls  along  then  at  full  speed,  at  10  yards  a  second  for 
20  seconds,  and  thus  goes  200  yards ;  then  the  brakes 
are  applied,  the  speed  is  slackened,  the  train  goes 
100  yards  in  20  seconds,  and  stops.  It  has  then  arrived 
at  the  next  station,  and  it  has  travelled  the  distance  in 
1  minute,  or  60  seconds. 

The  graph  (Fig.  96)  takes  all  these  circumstances  into 
account;  from  0  to  A  is  the  period  of  getting  up  speed 
(100  yards  in  20  seconds) ;  from  A  to  B,  the  period  of 
full  speed  (200  yards  in  20  seconds)  ;  and  from  B  to  M 
the  period  of  slackening  speed  until  finally  the  train  is 
brought  to  a  halt  (100  yards  in  20  seconds). 

It  is  sufficient  to  look  at  the  figure  to  realise  the  import- 
ance of  the  increasing  and  the  slackening  of  speed  over 
such  small  distances.  If  two  stations  were  distant 
from  each  other  200  yards  instead  of  400  yards,  the  period 
of  full  speed  would  completely  disappear,  and  it  would 
take  40  seconds  for  the  train  to  travel  200  yards. 

57.  Analytical  Geometry. 

The  general  idea  which  underlies  the  construction  of 
graphs  has  been  shown  in  section  46,  and  applied  under 
various  forms  in  the  pages  following.  It  consists,  as  we 
may  remember,  after  tracing  two  perpendicular  straight 
lines  OX,  OY,  in  setting  out  on  OX  a  length  x  =  OP, 
on  'OY  a  length  y  =  OQ,  and  determining  a  point  M 
by  drawing  through  P  and  Q  the  parallels  to  OY  and 
OX  which  cut  each  other  in  this  point  M. 


134  MATHEMATICS 

If  y  is  the  value  of  a  function  of  x  which  we  wish  to 
represent,  the  line  obtained  by  joining  all  the  points  M 
that  have  been  constructed  will  represent  the  variations 
of  the  function  y. 

By  means  of  some  new  illustrations,  we  are  going  to 
find  in  them  everything  which  is  at  the  base  of  an  impor- 
tant and  very  useful  science,  analytical  geometry, 
which  we  owe  to  the  genius  of  Descartes.1 

And  it  is  as  well  to  add  that  without  analytical  geometry 
we  never  could  have  imagined  graphs. 

The  two  straight  lines  OX,  OY  (Fig.  97)  are  called  the 
co-ordinate  axes  ;  OX  is  the  axis  of  the  x's,  or  the  axis 

of  the  abscissas,  OY  the  axis 
of  the  ?/'s,  or  the  axis  of  the 
ordinates. 

M  OP  =  x  and  OQ  =  y  are  the 
co-ordinates    of    the    point  M ; 
OP  is  the  abscissa  of  M,   and 


P          X  OQ  its  ordinate. 

A    negative    abscissa   would 
be    set    out    in    the    direction 
F      97  OX',    a   negative    ordinate    in 

the  direction  OY'. 

It  results  from  this  that  if  a  point,  as  seen  on  the 
figure,  is  in  the  angle  XOY,  its  x  and  its  y  are  positive  ; 
if  it  is  in  the  angle  YOX',  its  x  is  negative,  its  y  is  positive. 
X  OY',  its  x  and  its  y  are  negative ;  Y'OX,  its  x  is 
positive,  its  y  negative. 

If  a  point  is  marked  on  the  plane  of  the  figure,  we  then 
know  its  two  co-ordinates.  If  any  two  co-ordinates  are 
given,  we  know  the  position  of  the  corresponding  point. 

If  the  two  co-ordinates  a ,  y  are  not  simply  any  numbers, 
but  are  linked  by  an  algebraical  relation,  that  is  to  say, 
that  one  of  the  co-ordinates  being  known  the  other  may 
be  deduced  from  it  by  a  series  of  perfectly  definite  calcu- 

1  Rene  Descartes,  a  celebrated  philosopher  and  man  of  letters,  bora 
at  la  Have,  in  Touraine  (1596—1650). 


ANALYTICAL   GEOMETRY  135 

lations,  the  positions  of  M  will  lie  on  a  line.     The  alge- 
braical relation  in  question  is  the  equation  of  the  line. 

The  great  general  problems  with  which  analytical 
geometry  deals  are  : — 

1.  To  construct  a  line,  and   find   out  its  properties, 
knowing  its  equation  ; 

2.  To  find   the  equation  of  a  line,  when  it  has  been 
defined  in  a  precise  manner  by  any  means. 

Our  readers  need  not  be  ambitious  to  learn  what 
analytical  geometry  really  is.  But  in  constructing  our 
various  graphs  we  have  done  a  little  of  this  branch  of 
geometry  without  even  knowing  the  name  of  the  science  ; 
so  it  was  desirable  to  profit  by  the  occasion  given  us  to 
salute  in  passing  the  memory  of  one  of  the  greatest 
geniuses  of  whom  the  world  has  reason  to  be  proud. 

It  is  since  the  invention  of  Analytical  Geometry  that 
the  study  of  curved  lines  has  made  immense  progress, 
thanks  to  the  fresh  resources  which  this  science  has  brought 
to  bear  upon  them. 

Three  of  these  curved  lines,  however  (and  some  others 
also),  had  been  studied  in  antiquity  by  Greek  geometri- 
cians by  the  help  of  Geometry  alone.  The  mind  is 
absolutely  amazed  when  we  consider  what  power  of 
intellect,  what  prodigious  efforts  of  the  brain  have  been 
necessary  for  these  learned  men,  of  perhaps  more  than 
twenty  centuries  ago,  to  arrive  at  the  discoveries  by  which 
we  are  now  profiting. 

The  three  lines  of  which  we  are  going  to  speak  are 
to-day  in  continual  use,  even  in  practice.  For  this 
reason  we  have  resolved  to  say  something  about  them  in 
the  sections  which  follow,  not  to  study  them,  be  it  under- 
stood, but  simply  to  know  what  they  are,  so  that  the  pupil 
may  have  an  idea  of  the  pleasure  and  profit  he  will  have 
when,  later  on,  he  will  begin  to  take  them  seriously. 


136 


MATHEMATICS 


68.  The  Parabola. 

We  have  already  met  this  curve,  in  the  graphs  of  the 
falling  stone,  of  the  ball  tossed  up  in  the  air,  and  in  a 
portion  of  the  graph  of  the  underground  trains. 

The  precise  definition  of  the  parabola  is  (Fig.  98)  in 
that  each  of  its  points  M  is  equidistant  from  a  given  point 
F  and  from  a  given  straight  line 
(D),  so  that  MF  ==  MP.  The 
curve  then  takes  the  form  shown 
by  the  figure  ;  if  from  F,  which  is 
called  the  focus  of  the  parabola, 
a  perpendicular  line  is  lowered 
on  the  straight  line  (D)  called 
the  directrix,  this  straight  line 
FY  is  the  axis  of  the  curve, 
form  on  each  side  of  this  axis. 


FIG.  98. 
which    has    the 


same 


Tlie  axis  cuts  the  curve  at  A,  half-way  between  the  focus 
F  and  the  directrix.  The  point  A  is  the  apex  of  the 
parabola. 

If  AY  be  taken  for  the  axis  of  the  ordinates,  and  a 
perpendicular  AX  for  the  axis  of  the  abscissae,  the  equation 
of  the  parabola  would  be  y  —  kx*. 


59.  The  Ellipse. 

Many  of  the  arches  of  a  bridge  take  the  form  of  a  half- 
ellipse.  When  a  carrot  is  cut  obliquely  with  a  knife 
somewhat  regularly,  the  section  is  an  ellipse.  If  a  flat 
round  object  such  as  a  coin  is  held  up  against  a  lamp,  and 
the  shadow  thrown  on  a  piece  of  white  paper,  this  shadow 
may  also  be  an  ellipse. 

Astronomy  teaches  us  that  all  the  planets,  arid  ours  in 
particular,  turn  round  the  sun,  and  in  so  doing  describe 
ellipses. 

The   ellipse   is   determined   by   this   peculiar   quality' 


THE  ELLIPSE 


137 


that  the  sum  of  the  distances  of  any  of  its  points  from 
two  given  points  F,  F'  is  constant  ;  F,  F'  are  the  foci  of 
the  ellipse.  Let  us  suppose  we  wish  to  trace  an  ellipse 
on  a  sandy  soil.  This  can  be  done  by  fixing  two  pegs 
at  F,  F'  and  attaching  thereto  a  cord  (of  which  the  length 
has  been  given)  by  its  two  ends  ;  this  cord  is  held  out  by 
means  of  an  iron  spike  M  ;  if  this  spike  is  carried  over 
the  ground,  always  keeping  the  cord  stretched  out  tightly, 
it  will  trace  the  ellipse  ;  this  method  is  known  by  the  name 
of  "  the  gardeners'  mark." 

We  see  (Fig.  99)  that  the  ellipse  is  a  closed  curve  ; 
the  straight  line  AA'  is  called  the  focal  or  major  axis  ; 
the  middle  O   of    FF'   is    the 
cenlre.\   the  perpendicular  BB 
to  FF'  is  the  minor  axis  ;    the 
curve    has    a    form    exactly 
similar  above    and   below    the 
major  axis,  to  right  and  left  of 
the  minor  axis. 

The  major  axis  cuts  the  curve 
in  the  two  points  A,  A'  ;  the 
minor  in  B,  B  ;  the  4  points 
A,  A',  B,  B'  are  the  apexes 
of  the  ellipse.  It  is  easy  to  see  that  the  constant 
length  MF  +  MF'  is  equal  to  A  'A,  or  twice  OA  ;  this 
is  called  the  length  of  the  major  axis  ;  the  length  of  the 
minor  axis  is  BB',  or  twice  OB. 

If  the  two  points  F,  F'  were  to  become  one  alone,  in  O, 
then  the  ellipse  would  become  a  circle,  having  OA  =  OB. 

Taking  OA  and  OB  for  axes  of  the  #'s  and  the  ?/'s, 
the  equation  of  the  ellipse  would  be,  calling  a  the  length 
OA  and  b  the  length  OB, 


FIG.  99. 


The  equation  of  the  circle,  if  b  becomes  equal  to  a,  is 


l    = 


y      = 


188 


MATHEMATICS 


60.  The  Hyperbola. 

Although  this  curve  is  also  very  important,  it  is  not 
quite  so  easy  to  pick  out  ordinary  examples  of  it  as  in 
the  case  of  the  two  preceding  ones.  However,  if  a 
circular  lamp-shade  is  arranged  on  a  lamp,  and  then 
is  placed  in  such  a  way  that  the  light  is  below,  if  we 
look  at  the  shadow  which  is  cast  on  a  vertical  wall  by 
the  lower  edge  of  the  lamp-shade,  we  shall  see  a  fragment 
of  a  hyperbola. 

The  hyperbola   can  be  determined  by  the   following 
peculiar  quality  :  that  the  difference  of  the  distances  from 
any  one  of  its  points  to  two  fixed 
points   F,   F',   which   are   called 
foci,  is  constant. 

As  we  have  seen  just  now  for 
the  ellipse,  the  straight  line 
FF'  (Fig.  100)  and  the  perpen- 
dicular OY  raised  upon  the 
middle  of  FF  are  the  axes  oJ 
the  curve.  This  is  of  the  same 
form  both  above  and  below 
FF',  to  right  and  left  of  OY. 
the  curve  in  two  points  A,  A', 
FF'  is  called  a  transverse  axis  ; 
the  axis  OY  does  not  meet  the  curve.  The  segment  A'A 
has  a  length  equal  to  the  constant  difference  of  the 
distances  from  a  point  of  the  curve  to  F  and  to  F'. 

What  we  find  new  here  is  that  the  curve,  beside  being 
capable  of  being  extended  as  far  as  we  like,  is  made  up 
of  two  parts,  of  two  branches  as  one  may  say,  completely 
separated  one  from  the  other. 

We  must  note  the  existence  of  two  straight  lines  OC, 
OC',  which  are  called  the  asymptotes,  and  are  such  that, 
by  prolonging  them,  and  also  prolonging  the  curve,  we 
shall  see  the  curve  and  the  straight  line  approach  each 
other,  indefinitely,  without  ever  quite  running  into 


FIG.  100. 

The   axis    FF'   cuts 
which  are  the  apexes  ; 


THE  DIVIDED    SEGMENT  139 

one.  We  can  easily  construct  the  asymptotes,  knowing 
that  the  point  C  is  such  that  CA  is  perpendicular  to  FF' , 
and  that  OC  =  OF.  If  OA  =  a,  OC  =  c,  it  follows  that 
AC2  =  c2  —  a-  ;  supposing  that  AB  =  b,  and  taking 
OA,  OY  for  axes  of  the  #'s  and  the  ?/'s,  the  equation  of 
the  hyperbola  would  be 

?!  _  £  _  i 
a2       W  ~ 

What  we  must  specially  retain  in  our  minds  about  these 
very  condensed  remarks  on  the  three  very  important 
curves  about  which  we  have  just  been  speaking  is  that 
by  their  aid  many  and  various  constructions  may  be 
made,  and  also  that  they  contribute  to  the  acquisition  of 
that  manual  dexterity  which  is  so  necessary  in  tracing  all 
sorts  of  geometrical  curves.  For  this  purpose  the  pupil 
should  be  encouraged  to  use,  successively  or  alternatively, 
squared  paper,  the  usual  drawing  apparatus,  and  also 
outlines  in  freehand. 

61.  The  Divided  Segment. 

Let  AB  be  a  segment  of  a  straight  line ;  let  it  be  sup- 
posed that  it  is  produced  in  two  directions  (Fig.  101)  and 
that  M  be  a  moveable  point  on  the  straight  line  AB. 
If  the  point  M  is  placed,  for  example,  between  A  and  B,  it 
divides  AB  into  two  segments  AM,  MB,  and  it  is  the  ratio 

AM 
y  =  =-=fi  of  these  two  segments  that  we  wish  to  study. 

It  varies  evidently  according  to  the  position  of  M. 

We  will  place,  to  begin  with,  M  at  A  ;  the  ratio  is 
nothing,  since  MA  is  nothing  ;  if  M  is  moved  from  A  toward 
B,  the  ratio  becomes  greater ;  when  M  is  in  the  middle 
of  AB  the  ratio  y  is  equal  to  1  ;  when  M  is  brought  closer 
to  B,  y  has  values  which  become  greater  and  greater, 
and  it  is  said  that  when  M  arrives  at  B,  the  ratio  is  infinite  ; 
or  in  other  words,  it  is  so  enormously  large  that  it  cannot 
be  expressed  in  figures. 


MATHEMATICS 


If,    however,    M   passes   a    little    beyond    the    point 
B,  AM  will  be  always  positive,  MB  negative,  and  very 

AM 

small  ;    then  y,  that  is  to  say,       n,  will  be  negative  and 


very  large  ;    the  further  M  is  removed  from  B  (although 
the  ratio  will  remain  negative)   the  more  its  size  will 

t 


R* 

T~ 


FIG.  101. 

diminish,  remaining  always  greater  than  1,  but  approach- 
ing more  and  more  nearly  to  1. 

If  now,  beginning  with  the  point  M  at  A,  we  make  it 

AM 

move  towards  the  left,  the  ratio      &  becomes  once  more 


negative  ;  its  size  is  less  than  1,  and  it  approaches 
more  and  more  nearly  to  1  in  proportion  as  M  becomes 
distant  from  A. 

Representing,  for  each  position  of  the  point  M,  the 
value  of  the  ratio  y  by  an  ordinate  drawn  perpendicular 
to  the  straight  line  AB,  we  obtain,  as  a  graph  showing 
the  variations  of  this  ratio,  the  curve  seen  on  Fig.  101  ; 
this  curve  is  a  hyperbola,  of  which  the  asymptotes  are 
BY,  perpendicular  to  AB,  and  OX,  parallel  to  AB,  at  a 
distance  marked  by  the  unit,  and  below,  that  is  to  say, 
in  the  negative  direction. 

The   shape   of  the    figure    shows   that   there   are   not 

AM 

two  points  M  for  which  the  ratios    rU  can  be  the  same. 


DOH,  ME,  SOH;  GEOMETRICAL  HARMONIES    141 

As  soon  as  the  value  y  of  this  ratio  is  given,  with  its 
sign,  the  precise  position  of  M  is  determined  on  the 
straight  line  AB. 


62.  Doh,  me,  soh  ;  Geometrical  Harmonies. 

We  have   said   (Fig.  101)  that  there  cannot  exist  two 

AM 

such  different  points  M  that  the  ratio  is  the  same. 


But  a  point  M  being  given,  we  can  find  another  M'  from 

AM    AM' 

it,  and  only  one,  such  that  the  two  ratios  JTJTT>    *-r^f>  may 

have  the  same  size.     Since,  then,  the  signs  are  contrary, 

MA       AM 

we  have 


When  four  points  M',  A,  M,  B  are  such  that,  on  a 
straight  line,  they  may  exist  thus,  we  say  that  they  form 
a  harmonic  division. 

The  word  may  appear  strange.      Before  explaining  it 

..       M'A       AM 
we  are  going  to  write  the  proportion  ^TO  =  jrjg  rather 

differently  ;  let  us  call  the  segments  M'A,  M'M,  M'B, 
a,  m,  b.  Then  AM  =  m  —  a,  MB  =  b  —  m,  and  the 
relation  becomes 

a       m  —  a  .    m  —  a      b  —  m 

b  =  b~^n>  °r>  aSam>  "V        —b~~  ; 

m  m         (I        IN  1.12 

__I  =  I_F;  W^  +  _J=2;  _  +  _  =  _. 

On  the  other  hand,  when  we  begin  the  study  of  sound, 
we  learn  that  the  lengths  of  a  vibrating  chord  giving 
the  three  notes  doh,  me,  soh,  which  make  the  perfect 
major  chord,  are  proportional  to 

I  ^ 

II  5'  3' 


142  MATHEMATICS 

Then  the  inverse  lengths  are  proportional  to 


A»  4'  2' 

or  4,    5,    6  ; 

and,  as  4  +  6  =  2  x  5,  our  three  lengths  of  chords  a,  m, 
b  will  comply  with  the  relation 

-  +  -  -    - 

a,       b       nr* 

written  above. 

It   is    this   comparison   which   has   led   to   the   name 
"  harmonic  division." 

More  generally,  when  we  have  an  arithmetic  progression 
of  any  kind  whatever, 

a  b  c  ... 

and  1  is  divided  by  each  of  the  terms,  the  result 

111 

a   b    c 

thus  obtained  is  called  a  harmonic  progression. 

One  of  the  most  remarkable  properties  of  harmonic 
division,  and  one  which  plays  an 
important  part  in  geometry,  is  the 
following  : — 

Let  M'AMB  (Fig.  102)  be  a 
harmonic  division  ;  if  we  join  the 
four  points  which  compose  it  to 
any  point  P,  and  if  we  cut  the 
four  straight  lines  PM',  PA,PM,  PB 
by  any  straight  line  whatever,  we 
shall  still  have  a  harmonic  division. 

Thus  on  the  figure,  M'^MjBi,   M'2A2M2B2  are  harmonic 

divisions.     The  system  of  4  straight  lines  PM',  PA,  PM, 

PB  is  called  a  harmonic  sheaf  of  lines. 


FIG.  102. 


A  PARADOX :  64  =  65  148 


63.  A  Paradox :  65  =  65. 

In  mathematics  we  often  meet  with  paradoxes,  that 
is  to  say,  we  obtain  results  which  we  think  we  have 
worked  out  correctly,  which  are,  however,  obviously 
wrong. 

Any  paradox  unexplained  is  dangerous,  because  it 
throws  the  pupil's  mind  into  a  state  of  doubt  and 
confusion. 

When  a  paradox  is  explained,  on  the  contrary,  it  is 
instructive,  because  it  draws  attention  to  a  snare,  and 
shows  the  illusions  of  which  one  may  be  the  victim. 
Sometimes  it  is  incorrect  reasoning,  sometimes  it  is  a 
construction  too  loosely  made,  which  leads  to  a  flagrant 
absurdity. 

But  if  paradoxes,  properly  explained,  have  thus  their 
place  in  the  teaching  of  Geometry,  it  is  wise  to  adopt 
prudent  reserve  in  this  matter  in  elementary  instruction 
on  the  subject.  With  this  last,  of  course,  there  is  no 
question  of  going  deeply  into  things,  and  they  are  only 
indicated,  and  the  pupil  just  touches  them,  as  it  were, 
with  the  tip  of  his  finger. 

It  is  this  which  has  decided  me  to  refrain  up  to  now  from 
presenting  any  question  of  this  kind.  Having,  however, 
arrived  almost  at  the  end,  I  see  nothing  unwise,  indeed 
rather  the  contrary,  in  making  just  one  exception  which 
is  very  well  known  at  the  present  time.  This  we  might 
leave  the  pupil  to  seek  himself.  It  is  hardly  likely  he 
will  hit  upon  the  best  way,  and  it  will  be  best  to  come  to 
his  assistance  without  allowing  him  to  become  dispirited. 

We  will  take  (Fig.  103)  a  square  of  64  divisions  on  a 
piece  of  squared  paper  and  gum  this  on  cardboard.  This 
done,  the  lines  marked  on  the  figure  should  be  traced, 
and  the  square  will  be  found  to  be  split  up  into  two 
rectangles  having  8  sides  of  divisions  for  base,  and 
heights  of  5  and  3  sides ;  then  the  large  rectangle  will 


144 


MATHEMATICS 


be  split  up  into  two  trapeziums,  and  the  small  one  into 

two  triangles. 

Cut  up  the  cardboard    with  a  penknife  or  a  pair  of 

scissors,  by  following  the  three  traced  lines,  which  will 

then  give  us  the  four  pieces,  A,  B  (trapeziums)  and  C,  D 

(triangles). 

The  four  pieces  must  be  arranged  as  is  shown  in  the 
second  part  of  the  figure.  We  have 
a  rectangle  which  shows  5  columns 
of  13  divisions  each  ;  we  see  then 
5  X  13,  or  65  divisions,  with  this 
second  arrangement  ;  in  the  square 
there  will  be  only  8  x  8  or  64 
divisions.  These  two  different  results 
have  been  obtained  with  the  same 
pieces  of  cardboard.  This  is  enough 
to  make  us  imagine  that  our  heads 
have  become  bewildered,  seeing  that 
64  =  65. 

The  explanation  is  not  very  com- 
plicated once  it  is  put  plainly  before 
the  pupil,  but  it  needs  some  reflec- 
tion. Looking  at  the  long  diagonal 
of  the  rectangle,  in  the  second  part 
of  the  figure,  we  ask  ourselves  if  it  is 
really  a  straight  line.  It  is  made  up 
of  two  parts  :  the  hypotenuse  of 
the  rectangular  triangle  C,  and  the 

side  of  the  trapezium  A.     According  to  the  outline,  the 

Q 

slope  of  the  hypotenuse  on  the  large  side  is  ^  ;   that  of 

2 

the  side  of  the  trapezium  is  -z.     If  these  two  fractions 

were  exactly  equal,  we  should  have  a  straight  line.  But 
they  are  77;  and  ^ ;  the  first  is  a  little  less  than  the 
second,  and  what  appears  to  be  a  straight  line  is  really  a 


i 


FIG.  103. 


MAGIC  SQUARES  145 

quadrilateral,  very  thin  and  very  much  drawn  out,  which 
corresponds  to  the  area  of  the  added  division.  The 
union  seems  exact,  but  really  it  is  not  quite  perfect. 

If  we  took  a  square  of  21  x  21  =  441  divisions, 
dividing  the  side  into  13  and  8,  we  would  apparently 
have,  by  a  similar  construction,  441  =  442. 

In  that  case  the  two  fractions  whose  equality  would 

o 

be  necessary  to  make  a  perfect  match  would  be  ^i  and 

5  1 

=-^,  they  would  differ  only  by  ^oj  so  that  practically  the 

agreement  would  be  perfect. 

6*.  Magic  Squares. 

If  the  numbers  1  to  9  are  written  in  the  divisions  of  a 
square  in  the  following  manner, 

492 
357 
816 

we  can  prove  that,  on  adding  the  numbers  contained  in 
a  line,  in  a  column,  or  in  either  of  the  two  diagonals, 
the  result  is  always  the  same : — 4  +9+2=3+5+ 
7  =  8  +  1+6  =  4+3  +  8=9  +  5  +  1=2  +  7  + 
6  =  4  +  5  +  6  =  2  +  5  +  8  =  15. 

Such  a  figure  is  what  is  called  a  magic  square  of  3  ; 
the  sum  15  is  the  constant  magic  sum  ;  if  we  take  1  away 
from  each  figure,  which  then  reads 

381 

246 

705 

there  would  still  be  a  magic  square,  but  the  constant  would 
be  12  instead  of  15. 

Taking  the  numbers  0,  1,  2,  ...  24,  which  would  fill 
a  square  of  25  divisions  we  would  find  a  magic  square  of 
5 ;  the  constant  would  be  60. 

M.  L 


146  MATHEMATICS 

The  following  is  an  example  by  means  of  which  it  can 
be  proved  that  all  the  requisite  conditions  have  been 
properly  fulfilled  : — 

0  19  8  22  11 
23  12  1  15  9 
16  5  24  13  2 
14  3  17  6  20 

7  21   10     4  18 

and,  moreover,  if  the  square  is  cut  by  a  vertical  straight 
line  between  any  two  of  the  columns,  and  if  the  two  pieces 
are  interchanged,  we  still  have  a  magic  square.  Sup- 
posing that  the  square  be  cut  in  two  by  a  horizontal 
straight  line,  and  the  two  pieces  interchanged,  still  again 
we  find  a  magic  square. 

Ed.  Lucas  has  given  the  name  "  diabolical."  to  squares 
which  possess  this  property. 

Magic  squares  have  furnished  food  for  much  reflection. 
Although  they  appear  to  be  just  a  simple  game,  they  give 
rise  to  questions  which  present  great  difficulties,  and  even 
the  most  illustrious  mathematicians,  Fermat  amongst 
others,  have  not  disd  lined  to  occupy  themselves  with 
them.1 

We  can  hardly  ignore  the  existence  of  these  figures 
so  have  pointed  them  out  by  way  of  curiosity. 

65.  Final  Remarks. 

If  I  had  to  initiate  children  into  the  knowledge  of 
things   mathematically   essential,    such   as   we  have  dis 
cussed,  this  is  about  what  I  would  say  to  them  at  the  end 
oi'  our  course  : 

'•  You  are  going  to  begin  your  instruction  in  mathe- 
matical matters.  According  to  your  natural  dispositions, 

1  One  of  the  most  remarkable  works  published  on  this  question  in 
our  time  is  that  of  M.  G.  Arnoux  :  Arithm  tique  graphique  ;  Les  Espacea 
arithmdiques  hypermagiques  ;  Paris,  Gauthier-Villars,  1894. 


FINAL  REMARKS  147 

according  to  the  direction  which  you  will  be  called  upon 
to  follow  later  in  life,  this  instruction  will  be  more  or  less 
of  an  extended  nature  ;  but,  within  certain  limits,  it  will 
be  necessary  for  each  one  of  you. 

"  Up  to  the  present  you  have  studied  nothing,  but  you 
have  learnt  a  certain  number  of  useful  things,  by  way  of 
amusement.  If  you  have  made  any  effort,  it  has  been 
purely  a  voluntary  one  on  your  part,  nothing  has  been 
required  from  you,  and,  particularly,  nothing  from  your 
memory. 

"  Before  knowing  how  to  read  or  write,  you  have  been 
able  to  make  up  numbers  with  the  aid  of  various  objects, 
and  to  do  several  simple  problems.  When  it  has  been 
possible  to  employ  figures,  the  practice  of  calculation  has 
become  more  easy  for  you.  Thanks  to  the  custom  of 
carrying  you  back  to  the  objects  themselves,  and  of  not 
only  considering  the  figures  which  translate  them,  you 
have  very  early  arrived  at  the  idea  of  negative  numbers, 
and  become  quite  familiar  with  it.  Some  notions  of 
geometry,  found  out,  but  not  demonstrated,  have  been 
sufficient  to  begin  to  make  you  see  the  close  bond  which 
unites  the  science  of  numbers  to  that  of  space. 

"You  have  not  made  a  study  of  fractions  more  than 
any  other  study,  but  you  know  what  a  fraction  is,  and 
you  have  a  fair  grasp  of  the  calculations  which  belong 
to  it. 

"  By  progressions,  first  in  simple  form,  then  somewhat 
more  generalised,  you  have  been  led  to  the  idea  of 
enormous  numbers.  Other  large  numbers  appeared 
before  your  eyes  when  you  saw  what  a  permutation 
meant. 

"  With  some  practical  notions  of  geometry  and  drawing 
at  the  same  time  you  have  succeeded  in  grasping  the 
construction  and  the  use  of  graphs,  and  applying  your 
knowledge  especially  to  questions  of  movement.  You 
have  thus  arrived,  as  it  were,  at  the  door  of  analytical 
geometry;  you  have,  at  least,  perceived  the  form  of  the 

L2 


148  MATHEMATICS 

three  principal  curves  that  analytical  geometry  permits 
us  to  study  more  deeply,  but  which  the  ancients  already 
knew. 

"  Whether  of  all  these  ideas  much  or  little  remains  in 
your  memory,  you  are  certain  to  have  retained  something. 
You  have  at  the  same  time  acquired,  without  any  doubt, 
certain  habits  of  mind  which  are  now  going  to  prove  of  the 
utmost  value  to  you. 

"  Henceforward  you  have  not  to  do  with  play  but  with 
work.  You  ought  to  subject  yourself  to  intellectual 
efforts,  perhaps  also  to  some  efforts  of  memory.  They 
will  be  the  less  formidable  because  up  to  now  your  forces 
have  been  husbanded,  and  you  know  many  more  things 
than  other  children  of  your  age  who  have  been  subjected 
to  a  sort  of  torture,  that  of  forcing  them  to  retain  words  hi 
their  minds  without  understanding  anything  about  them. 

"  In  the  majority  of  the  objects  of  your  studies  in  the 
future  you  will  find  things  cropping  up  that  you  knew  of 
old ;  any  trouble  which  novelty  brings  to  you  will  be 
soon  wiped  out.  Do  not  think,  however,  that  you  will 
never  meet  any  difficulties  ;  you  will  find  them,  but  know- 
ing that  they  are  only  in  the  nature  of  things,  that  it  is 
necessary  to  surmount  them  in  order  to  arrive  at  interesting 
and  useful  results,  you  will  find  that  you  possess  the  neces- 
sary courage.  In  play  you  have  acquired  ideas,  and  your 
studies  in  future  will  thereby  be  facilitated.  In  your  work 
henceforward  you  are  going  to  make  the  most  of  what  you 
know;  you  will  exercise  your  reason;  you  will  augment 
the  extent  of  your  knowledge.  But  this  work,  even  if  it 
is  no  longer  play,  will  not  prove  to  be  a  burden !  You 
will  find  a  pleasure  in  it,  knowing  it  to  be  useful ;  little 
by  little  it  will  become  a  necessity  of  your  life ;  it  will  not 
only  be  easy,  but  necessary. 

"  In  case  of  doubt,  besides,  you  will  have  teachers  who 
will  be  guides  to  you  ;  but  do  not  ask  anything  more  from 
them.  Personal,  untrammelled  effort  can  alone  give  good 
results.  You  have  unconsciously  acquired  the  habit  ID 


FINAL  REMARKS  140 

the  games  of  your  childhood.  Now  it  is  for  you  to  make 
the  most  of  it  by  bringing  to  the  task  of  acquiring  know- 
ledge all  the  patience,  the  energy,  the  determination  that 
you  have  held  in  reserve  !  :' 

Such  is  about  the  substance  of  what  ought  to  be  said 
to  the  child  at  the  end  of  these  introductory  occupations, 
on  the  eve  of  undertaking  his  studies.  To  make  him 
grasp  these  ideas  you  must  not  deliver  a  lecture,  but  you 
must  explain  it,  if  necessary,  in  ten  or  twenty  talks.  The 
teacher  will  have  to  draw  from  them  the  material  to  light 
the  pupil  along  the  new  path  that  he  is  called  upon  to 
follow. 

The  introductory  process,  to  my  mind,  ought  to  be 
specially  carried  out  in  the  home.  But  even  when,  from 
any  reason  whatever,  personal  or  social,  this  cannot  be,  the 
father  and  mother  ought  to  remember  that  their  first  duty 
is  to  associate  themselves  with  the  evolution  of  the  child's 
brain,  and  to  be  at  any  rate  a  help  to  the  teacher,  even  if, 
from  any  cause,  they  themselves  have  been  unable  to  nil 
the  post. 

And  as,  once  the  introductory  stage  passed,  that  of 
instruction  begins,  the  duty  of  the  parents  becomes  more 
important  still  (if  that  is  possible) ;  their  responsibility  is 
heavy,  for,  whether  for  good  or  evil,  the  whole  destiny  of 
their  child  may  be  influenced,  according  to  the  decision 
of  the  father  and  mother. 

It  is  to  these  that  I  turn  my  attention  for  the  moment, 
to  give  a  few  words  of  advice — in  my  opinion,  at  least, 
good  advice — of  which  each  can  take  any  portion  that  is 
likely  to  be  useful. 

To  begin  with,  we  all  agree  on  one  point — that  an 
introduction  to  the  science  of  mathematics  is  indispensable 
to  any  child,  without  distinction  of  fortune,  of  social 
position,  or  of  sex ;  but  I  also  maintain  that,  without  any 
distinction  or  reserve,  mathematical  instruction  is  equally 
indispensable. 

Women  have  need  of  it  just  as  much  as  men  ;  every-day 


150  MATHEMATICS 

life,  domestic  economy,  no  less  than  the  manufactures 
and  arts  whose  applications  have  to  do  with  our  existence, 
require  from  us  all  a  knowledge  of  the  science  of  size 
and  space. 

Here  an  objection  presents  itself  which  I  have  refuted 
a  hundred  times  already,  but  will  discuss  once  more 
with  my  readers.  Parents  say  to  me,  "  Has  my 
child  any  gift  for  mathematical  study  ?  If  he  is  not  so 
gifted,  is  it  not  losing  his  time  to  direct  his  studies  in 
that  particular  channel  ?  We  do  not  intend  to  make  him 
into  a  mathematician." 

This  is  all  very  well.  But  when  you  taught  the  same 
child  reading  and  writing,  did  you  ask  yourself  whether 
he  had  any  gift  for  these  branches  of  study  ?  When  you 
inculcated  the  first  principles  of  drawing,  did  you  think 
he  was  intended  to  become  a  great  painter  ?  No  one 
doubts  the  necessity  that  exists  for  each  man  and  woman 
to  learn  how  to  express  his  or  her  ideas  correctly  in  the 
mother  tongue  ;  and  when  that  is  achieved,  surely  we 
do  not  imagine  that  each  of  them  is  destined  to  become 
a  Shakespeare  or  a  Milton. 

No  more  in  mathematics  than  in  other  subjects  does 
instruction  make  learned  men ;  there  is  no  question 
of  making  them  ;  but  there  exists  in  everything  a  general 
groundwork  of  useful  knowledge,  which  is  necessary  and 
at  the  same  time  easy  for  everybody  to  acquire  whose 
brain  is  not  in  any  way  defective. 

The  whole  of  this  knowledge  on  various  subjects  can 
be  acquired,  thanks  to  the  preliminary  introduction,  in 
much  less  time  than  is  given  up  to  it  in  the  ordinary 
course  of  teaching. 

This  literary  knowledge,  as  far  as  our  subject  is  con- 
cerned, is  pretty  nearly  represented  by  elementary 
mathematics.  Any  child,  whether  gifted  or  not  in  any 
special  manner,  can  assimilate  the  whole  of  this  knowledge, 
just  as  he  can  learn  to  read  and  write  correctly,  if  not 
elegantly.  If  he  has  an  inborn  taste  for  mathematics, 


FINAL   REMARKS  151 

he  will  continue  his  studies  in  that  direction ;  if  he  is 
literary  by  temperament,  he  will  write.  Teaching  has 
never  made  learned  men  or  artists,  its  aim  should  be  the 
preparation  of  men's  minds. 

Then  let  there  be  no  hesitation  on  this  point.  Your 
child  ought  to  acquire  the  fundamental  notions  of  mathe- 
matics necessary  for  everybody. 

We  should  always  bear  in  mind  the  apt  and  suggestive 
remark  of  M.  Emile  Borel : 

''A  mathematical  education  at  once  theoretical  and 
practical  can  exercise  the  happiest  influence  over  the 
formation  of  the  m'.nd." 

I  am  content  to  leave  you  under  this  impression. 


INDEX 


ABSCISSAE,  134 
Abstract  numbers,  5 
Acceleration,  129 
Acute  angle,  53 
Addition,  9,  21,  24 

of  fractions,  51 
Algebra,  26 
Altitude,  55 

Analytical  geometry,  133 
Angles,  53,  54 
Apex,  54,  56,  59 

of  hyperbola,  138 

parabola,  136 
Arc   104 
Archimedes,  111 
Area,  60,  106 

Arithmetical  progressions,  89 
triangles        and 

squares,  80 
Arnoux,  146 
Asymptote,  138 
Axes,  co-ordinate,  134 
Axis  of  ellipse,  137 

hyperbola,  138 

parabo'a,  136 


BASE,  55,  56,  57,  58,  etc. 
of  numeration,  82 
Binary  numeiation,  87 
Borel,  In 2 
Brackets,  27 
Bundles,  5 


CARRYING,  21 
Centre  of  circle,  104 
ellipse,  137 
Chessboard  problems,  81,  93, 

100 

Chord,  104 
Circle,  104,  106,  110 
Circumference,  105 
Coloured  counters,  17 
Common  difference,  90 

ratio,  91 
Compass,  102 
Composite  numbers,  40 
Compound  interest,  95 
Concrete  numbers,  5 
Concurrent,  54 
Cone,  110 
Convex,  56 
Co-ordinates,  134 
Counting,  7,  82 
Cube,  38,  59,  68 
Cylinder,  110 


DECAGON,  56 
Decimal  fraction,  47 
system,  82 
Degree,  103 
Denominator,  45,  49 
Deprez,  84 
Descartes,  104 
Diabolical  squares,  145 


154 


INDEX 


Diagonal,  56 
Diameter,  104 
Difference,  12,  90 
Directrix,  136 
Dividend,  42 
Division,  42 
Divisor,  42 
Dodecagon,  56 
Drawing,  2 
Duodecimal  system,  82 


EDGE,  59 
Ellipse,  136 
Equal,  26 
Equation,  105 
Equilateral  triangle,  54 
Eratosthenes,  42 
Even  number,  8 


FACE,  59 
Factor,  36,  42 
Factorial,  98 
Faggots,  5 
Falling  stone,  129 
Fermat,  81 
Figures,  19 
Focal  axis,  137 
Focus,  136 
Fractions,  45,  49 
Fro3bel,  3 
Function,  113 


GEOMETRICAL  progression,  91 
Geometry,  52 

analytical,  133 
Godard,  85 
Grade,  103 
Graphs,  112—128,  135,  etc. 


Great  circle,  110 
Guillaume,  101 


HANOI  tower  puzzle,  88 
Harmonic  division,  141 

sheaf  of  lines,  142 
Height,  55,  56,  58 
Heptagon,  56 
Hexagon,  56 
Hippocrates,  107 
Horizontal,  53 
Hundred,  7 
Hyperbola,  138 
Hypotenuse,  64 


IMPROPER  fraction,  49 
Index,  38 

Initial  velocity,  130 
Inscribed  angle,  104 
Interest,  95 
Investment,  95 
Isosceles  triangle,  54 


LA  CHALOTAIS,  3 
Leibnitz,  87 
Lucas,  47,  etc. 


MAGIC  squares,  145 

Major  axis,  137 

Matches,  5 

Mathematics,  importance    of, 

150 
teaching        of, 

151 

Measures,  31 
Mental  calculation,  8 


INDEX 


155 


Metric  system,  32 
Millions,  14 
Minor  axis,  137 
Minus,  26,  28 
Mohammedan  method,  37 
Multiplicand,  35 
Multiplication,  33,  35 

of     fractions, 

51 
Multiplicator,  35 

NEGATIVE  numbers,  28,  30 
Nought,  20,  87 
Numbers,  5 
Numeration,  8 

systems  of,  82 
Numerator,  45,  49 

OBTUSE  angle,  53 
Obtuse-angled  triangle,  54 
Octagon,  56 
Odd  numbers,  8 
Ordinate,  134 

PARABOLA,  129,  136 
Paradox,  143 
Parallel,  52 
Parallelogram,  57 — 63 
Parallelepiped,  59 
Pascal,  80 
Pentagon,  56 
Permutations,  97 
Perpendicular,  53 
Pestalozzi,  3 

Pictured  permutations,  98 
Pile,  75 
Plane,  52 
Plus,  26 

Polygon,  65 — 63 
Polyhedra,  109 
Positive  numbers,  28 


Powers,  38,  101 

of  eleven,  79 

Prime  numbers,  40 

Prism,  59 

Product,  35,  40 

Progressions,  89 

arithmetical,  89 
decreasing,  92 
geometric,  91 
harmonic,  142 
increasing,  92 

Proper  fraction,  49 

Proportion,  31,  51 

Protractor,  102 

Puzzles,  88,  etc. 

Pyramid,  59 

Pythagoras,  33 

QUADRILATERAL,  56 
Quotient,  42,  91 

RADIUS,  104 
Ratio,  139 

Rectangle,  48,  57,  63 
Re-entering  angle,  56 
Remainder,  12,  44 
Rhombus,  57 
Right  angle,  53 
Right-angled  triangle,  54 
Ring  puzzle,  88 
Roman  numeration,  85 
Rose,  107 

SECANT,  54,  105 
Segment,  24 

of  circle,  104 
Semicircle,  107 
Side,  54 
feigc,  28,  44 
Snail  problem,  120 
Sphere,  110 


156 


INDEX 


Square,  38,  57,  61,  63 

arithmetical,  80 
numbers,  72 

Squared  paper,  1,  33,  48,  61 
Sticks,  5 

Straight  line,  23,  52 
Strokes,  1 
Subtend,  104 
Subtraction,  12,  22,  25 

of  fractions,  51 
Sum,  10 

of  arithmetical  progres- 
sion, 90 
cubes,  76 
geometrical   progres 

sion,  92 
squares,  75 
Systems  of  numeration,  82 

TANGENT,  105 
Ten,  6 
Term,  45,  90 


Thousand,  14 
Total,  10 

Transverse  axis,  138 
Trapezium,  57,  63,  91 
Travelling    problems,     115 — 

128,  etc. 
Triangle,  54,  63 

arithmetical,  SO 
Triangular  numbers,  70 

UNDERGROUND  trains,  132 
Unit,  61 
Unity,  31 

VERTICAL,  53 
Viete,  31 
Volume,  109 

WEATHER  grapl*,  121 
Weighing,  84 

ZERO.  20 


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